WEBVTT 00:00:00.000 --> 00:00:02.490 align:middle line:90% [SQUEAKING] 00:00:02.490 --> 00:00:04.482 align:middle line:90% [RUSTLING] 00:00:04.482 --> 00:00:06.474 align:middle line:90% [CLICKING] 00:00:06.474 --> 00:00:18.930 align:middle line:90% 00:00:18.930 --> 00:00:20.790 align:middle line:90% CASEY RODRIGUEZ: All right. 00:00:20.790 --> 00:00:26.580 align:middle line:84% So last lecture, we concluded our discussion 00:00:26.580 --> 00:00:30.900 align:middle line:84% about measurable functions, I mean, measurable sets. 00:00:30.900 --> 00:00:33.750 align:middle line:84% And remember our original motivation was 00:00:33.750 --> 00:00:38.590 align:middle line:84% that we're trying to build an integral that somehow surpasses 00:00:38.590 --> 00:00:44.240 align:middle line:84% the properties of that of the Riemann integral in that, 00:00:44.240 --> 00:00:47.030 align:middle line:84% hopefully, this larger class of functions 00:00:47.030 --> 00:00:49.430 align:middle line:84% that are integral with respect to this integral 00:00:49.430 --> 00:00:55.830 align:middle line:84% form a Banach space as opposed to Riemann integral functions. 00:00:55.830 --> 00:01:00.430 align:middle line:84% And we started off by asking the question, 00:01:00.430 --> 00:01:04.290 align:middle line:84% how would we integrate the simplest types of functions, 00:01:04.290 --> 00:01:07.080 align:middle line:84% which are just one on a set and zero off of it. 00:01:07.080 --> 00:01:13.830 align:middle line:84% And that led us to how to define measure. 00:01:13.830 --> 00:01:19.490 align:middle line:84% And so then we define Lebesgue measurable sets, 00:01:19.490 --> 00:01:22.580 align:middle line:84% proved that they form this special type of collection 00:01:22.580 --> 00:01:24.020 align:middle line:84% of sets called the sigma algebra, 00:01:24.020 --> 00:01:27.440 align:middle line:84% and that they contain a lot of sets, 00:01:27.440 --> 00:01:31.970 align:middle line:84% a lot of interesting sets, open, closed, unions, 00:01:31.970 --> 00:01:34.340 align:middle line:84% countable unions of closed, which are not necessarily 00:01:34.340 --> 00:01:37.730 align:middle line:84% closed, countable intersections of opens, 00:01:37.730 --> 00:01:39.770 align:middle line:84% which are not necessarily open, and so on. 00:01:39.770 --> 00:01:42.590 align:middle line:90% 00:01:42.590 --> 00:01:45.910 align:middle line:84% Now not every set is Lebesgue measurable. 00:01:45.910 --> 00:01:47.650 align:middle line:84% We will not go through that construction 00:01:47.650 --> 00:01:50.620 align:middle line:90% of a nonmeasurable set. 00:01:50.620 --> 00:01:57.190 align:middle line:84% The construction of the set or how one obtains it 00:01:57.190 --> 00:02:03.810 align:middle line:84% also would provide a proof that we could not 00:02:03.810 --> 00:02:07.080 align:middle line:84% build a measure that's defined on all subsets of real numbers 00:02:07.080 --> 00:02:11.670 align:middle line:84% having these three properties that is translation invariant. 00:02:11.670 --> 00:02:14.760 align:middle line:84% The measure of an interval is the length of the interval. 00:02:14.760 --> 00:02:17.730 align:middle line:84% And the measure of a disjoint union 00:02:17.730 --> 00:02:19.650 align:middle line:90% is the sum of the measures. 00:02:19.650 --> 00:02:23.880 align:middle line:84% But this is a functional analysis class. 00:02:23.880 --> 00:02:28.120 align:middle line:84% Our goal is to build up a Banach space of integral functions 00:02:28.120 --> 00:02:30.450 align:middle line:84% so we needed to define a better notion of integral 00:02:30.450 --> 00:02:32.430 align:middle line:90% than Riemann. 00:02:32.430 --> 00:02:34.920 align:middle line:84% I imagine in a measured theory class, 00:02:34.920 --> 00:02:36.480 align:middle line:84% you would see such a construction 00:02:36.480 --> 00:02:39.320 align:middle line:90% but we won't cover that here. 00:02:39.320 --> 00:02:42.010 align:middle line:84% So in some sense, if we've built up measure, 00:02:42.010 --> 00:02:48.440 align:middle line:84% we kind of know, roughly speaking, 00:02:48.440 --> 00:02:50.210 align:middle line:84% how we would integrate the simplest 00:02:50.210 --> 00:02:54.050 align:middle line:84% type of function, which is one on a measurable set 00:02:54.050 --> 00:02:56.180 align:middle line:90% and zero off of it. 00:02:56.180 --> 00:02:59.450 align:middle line:84% Now what about what would be a method 00:02:59.450 --> 00:03:03.630 align:middle line:84% of trying to integrate more general functions? 00:03:03.630 --> 00:03:07.340 align:middle line:90% So this is, I think-- 00:03:07.340 --> 00:03:13.590 align:middle line:84% so now we're going to be talking about measurable-- 00:03:13.590 --> 00:03:17.080 align:middle line:90% this is how it's spelled-- 00:03:17.080 --> 00:03:18.580 align:middle line:90% measurable functions. 00:03:18.580 --> 00:03:26.800 align:middle line:90% 00:03:26.800 --> 00:03:30.270 align:middle line:84% So to motivate the definition of a measurable function, 00:03:30.270 --> 00:03:36.000 align:middle line:84% let me just give a few minutes of why 00:03:36.000 --> 00:03:39.000 align:middle line:84% we introduced this definition, what's really behind it. 00:03:39.000 --> 00:03:41.610 align:middle line:84% So historically, when Lebesgue thought 00:03:41.610 --> 00:03:46.920 align:middle line:84% of his theory of integration, if we had a function on a closed 00:03:46.920 --> 00:03:48.030 align:middle line:90% interval, AB-- 00:03:48.030 --> 00:03:49.590 align:middle line:90% and let me draw it. 00:03:49.590 --> 00:03:55.770 align:middle line:90% Let's say it's just increasing. 00:03:55.770 --> 00:04:00.500 align:middle line:84% So what Riemann does, of course, is you partition up ab. 00:04:00.500 --> 00:04:08.850 align:middle line:84% And then you form, essentially, these boxes 00:04:08.850 --> 00:04:12.450 align:middle line:84% that have width given by how you've chopped up 00:04:12.450 --> 00:04:17.190 align:middle line:84% ab and height given by f, the function f, evaluated 00:04:17.190 --> 00:04:20.200 align:middle line:84% at some point in those subintervals. 00:04:20.200 --> 00:04:20.700 align:middle line:90% Right. 00:04:20.700 --> 00:04:22.870 align:middle line:84% And you form a Riemann sum and you take a limit. 00:04:22.870 --> 00:04:23.370 align:middle line:90% Yeah. 00:04:23.370 --> 00:04:25.110 align:middle line:84% And that gives you the Riemann integral. 00:04:25.110 --> 00:04:31.490 align:middle line:84% And so what Lebesgue thought of doing was, instead of 00:04:31.490 --> 00:04:35.610 align:middle line:84% chopping up the domain, chopping up the range. 00:04:35.610 --> 00:04:44.540 align:middle line:84% So let's imagine-- so this function is only this height. 00:04:44.540 --> 00:04:49.660 align:middle line:84% So what he would do, or what one would do, is let's-- 00:04:49.660 --> 00:04:51.970 align:middle line:84% and let's say we're just interested in integrating 00:04:51.970 --> 00:04:53.710 align:middle line:90% non-negative things-- 00:04:53.710 --> 00:05:04.385 align:middle line:84% chop up the range and so on up to some finite point. 00:05:04.385 --> 00:05:07.660 align:middle line:90% 00:05:07.660 --> 00:05:12.220 align:middle line:84% And now how do you form your boxes, if you like, 00:05:12.220 --> 00:05:16.310 align:middle line:84% that you're going to take the size of? 00:05:16.310 --> 00:05:23.720 align:middle line:84% Well, you look at the piece of f that's 00:05:23.720 --> 00:05:27.890 align:middle line:90% in between a given partition. 00:05:27.890 --> 00:05:32.790 align:middle line:90% And this portion of f will-- 00:05:32.790 --> 00:05:33.290 align:middle line:90% OK. 00:05:33.290 --> 00:05:40.965 align:middle line:84% In this picture, we'll call this point c. 00:05:40.965 --> 00:05:48.770 align:middle line:90% 00:05:48.770 --> 00:05:51.830 align:middle line:84% So then you take this portion of f that's in here. 00:05:51.830 --> 00:05:56.060 align:middle line:84% And then ac would be given by is the set f inverse of, 00:05:56.060 --> 00:05:57.410 align:middle line:90% in this picture-- 00:05:57.410 --> 00:06:01.310 align:middle line:84% and I'm just going to write it in this way-- 00:06:01.310 --> 00:06:05.720 align:middle line:90% yi, y, yi minus 1, yi. 00:06:05.720 --> 00:06:11.260 align:middle line:90% 00:06:11.260 --> 00:06:15.730 align:middle line:84% And you could build up a box by now 00:06:15.730 --> 00:06:23.140 align:middle line:84% taking it to be the lower value y2 00:06:23.140 --> 00:06:27.430 align:middle line:84% and having width given by a to c. 00:06:27.430 --> 00:06:29.110 align:middle line:84% And I can actually do this because this 00:06:29.110 --> 00:06:33.130 align:middle line:84% is an increasing function so this will be an actual box. 00:06:33.130 --> 00:06:37.890 align:middle line:90% Now what am I getting at? 00:06:37.890 --> 00:06:40.860 align:middle line:84% And then you would say something like we 00:06:40.860 --> 00:06:48.330 align:middle line:84% would like to define the integral of f over ab. 00:06:48.330 --> 00:06:52.020 align:middle line:84% So this is all motivation, informal discussion. 00:06:52.020 --> 00:06:55.450 align:middle line:84% Don't take this to heart too much. 00:06:55.450 --> 00:06:57.780 align:middle line:84% Then we would like to define the integral from ab 00:06:57.780 --> 00:07:00.537 align:middle line:90% to f to be somehow this limit. 00:07:00.537 --> 00:07:02.370 align:middle line:84% And I'm not even going to write as something 00:07:02.370 --> 00:07:04.830 align:middle line:84% goes to zero or infinity you can think of 00:07:04.830 --> 00:07:18.200 align:middle line:84% as partitions get smaller of a sum now where I'm summing 00:07:18.200 --> 00:07:27.740 align:middle line:84% from i equals 1 to n of y minus 1, so the lower part, 00:07:27.740 --> 00:07:34.280 align:middle line:84% times now the length of, at least for this picture-- 00:07:34.280 --> 00:07:38.810 align:middle line:84% so this would be the length of the interval given 00:07:38.810 --> 00:07:46.780 align:middle line:90% by f inverse of yi minus 1, yi. 00:07:46.780 --> 00:07:48.530 align:middle line:90% OK? 00:07:48.530 --> 00:07:55.160 align:middle line:84% So this would be kind of an analogous procedure 00:07:55.160 --> 00:07:57.470 align:middle line:84% to what Reimann does on the domain 00:07:57.470 --> 00:07:59.240 align:middle line:84% except now focusing on the range. 00:07:59.240 --> 00:08:01.370 align:middle line:84% Now I wrote length here because this 00:08:01.370 --> 00:08:05.960 align:middle line:84% function I drew is increasing so the inverse image of one 00:08:05.960 --> 00:08:08.280 align:middle line:84% of these intervals is going to be another interval. 00:08:08.280 --> 00:08:11.270 align:middle line:90% So the length is meaningful. 00:08:11.270 --> 00:08:23.550 align:middle line:84% But if f is more general, f inverse 00:08:23.550 --> 00:08:45.810 align:middle line:84% doesn't need to be of yi minus 1, yi, need not be an interval. 00:08:45.810 --> 00:08:49.940 align:middle line:84% And so taking its length would not be a meaningful thing. 00:08:49.940 --> 00:08:55.810 align:middle line:84% But remember we now have this notion of measure, which 00:08:55.810 --> 00:08:58.570 align:middle line:84% should be the substitute for length for more 00:08:58.570 --> 00:09:00.760 align:middle line:90% general sets, measurable sets. 00:09:00.760 --> 00:09:04.420 align:middle line:84% So this procedure could still work 00:09:04.420 --> 00:09:08.500 align:middle line:84% if, instead of this requiring that this is an interval, 00:09:08.500 --> 00:09:12.370 align:middle line:84% requiring that this is a Lebesgue measurable set. 00:09:12.370 --> 00:09:14.980 align:middle line:84% And then perhaps one could define the integral 00:09:14.980 --> 00:09:16.960 align:middle line:90% in this way. 00:09:16.960 --> 00:09:20.745 align:middle line:84% And there these kind of things would no longer be boxes. 00:09:20.745 --> 00:09:26.120 align:middle line:90% 00:09:26.120 --> 00:09:29.600 align:middle line:84% So that should motivate why perhaps we 00:09:29.600 --> 00:09:32.840 align:middle line:84% should look at functions so that the inverse image 00:09:32.840 --> 00:09:37.640 align:middle line:84% of closed intervals is a measurable set so that we can 00:09:37.640 --> 00:09:40.700 align:middle line:84% take its measure and maybe do this procedure of defining 00:09:40.700 --> 00:09:41.510 align:middle line:90% an integral. 00:09:41.510 --> 00:09:44.760 align:middle line:90% Now all of that is a bit-- 00:09:44.760 --> 00:09:46.580 align:middle line:84% again, this is all informal discussion 00:09:46.580 --> 00:09:49.460 align:middle line:90% just meant to be motivation. 00:09:49.460 --> 00:09:54.290 align:middle line:84% In fact, in how we'll define the Lebesgue integral, 00:09:54.290 --> 00:09:56.240 align:middle line:84% we're not going to define it in this way, 00:09:56.240 --> 00:10:01.400 align:middle line:84% because what this way of doing it suffers 00:10:01.400 --> 00:10:06.970 align:middle line:84% is it's not clear that this is independent of how 00:10:06.970 --> 00:10:10.790 align:middle line:90% I partitioned up the range. 00:10:10.790 --> 00:10:13.870 align:middle line:84% So maybe if I take a limit as the partitions get smaller 00:10:13.870 --> 00:10:16.450 align:middle line:84% along some sequence of partitions, 00:10:16.450 --> 00:10:18.200 align:middle line:84% I get a different number from the others 00:10:18.200 --> 00:10:20.200 align:middle line:84% so that would have to be checked. 00:10:20.200 --> 00:10:23.560 align:middle line:84% But as we'll see when we do define the Lebesgue 00:10:23.560 --> 00:10:28.210 align:middle line:84% integral, this number, you can compute it 00:10:28.210 --> 00:10:31.840 align:middle line:84% by essentially this procedure I gave here 00:10:31.840 --> 00:10:39.310 align:middle line:84% where you chop up the range of f and take approximations 00:10:39.310 --> 00:10:43.000 align:middle line:84% to f that are more general than just step functions, which 00:10:43.000 --> 00:10:44.320 align:middle line:90% is what this is. 00:10:44.320 --> 00:10:49.270 align:middle line:84% It's a function that's a given value off on an interval 00:10:49.270 --> 00:10:50.290 align:middle line:90% and then zero outside. 00:10:50.290 --> 00:10:52.880 align:middle line:90% 00:10:52.880 --> 00:10:59.660 align:middle line:84% And that'll be a way of seeing that this motivation connects 00:10:59.660 --> 00:11:02.570 align:middle line:84% to actually how we define the Lebesgue integral. 00:11:02.570 --> 00:11:09.280 align:middle line:84% But again, the whole point of what I'm saying is that-- 00:11:09.280 --> 00:11:12.380 align:middle line:84% we're trying to motivate is that we should, 00:11:12.380 --> 00:11:13.910 align:middle line:84% and at least historically this is 00:11:13.910 --> 00:11:17.850 align:middle line:84% why one considers these things, consider functions 00:11:17.850 --> 00:11:23.380 align:middle line:84% so that the inverse image of closed intervals is measurable. 00:11:23.380 --> 00:11:25.800 align:middle line:90% OK. 00:11:25.800 --> 00:11:30.430 align:middle line:84% And that's the motivation for measurable functions. 00:11:30.430 --> 00:11:38.690 align:middle line:84% Now, as we saw when we were discussing 00:11:38.690 --> 00:11:43.440 align:middle line:84% what sets are measurable, we didn't exactly 00:11:43.440 --> 00:11:48.660 align:middle line:84% conclude directly that closed intervals were measurable. 00:11:48.660 --> 00:11:55.060 align:middle line:84% We started with something a little more basic, 00:11:55.060 --> 00:11:57.647 align:middle line:84% which was half infinite intervals, 00:11:57.647 --> 00:11:59.230 align:middle line:84% proved those were measurable, and then 00:11:59.230 --> 00:12:02.890 align:middle line:84% concluded that close intervals are measurable. 00:12:02.890 --> 00:12:04.910 align:middle line:90% Any interval is measurable. 00:12:04.910 --> 00:12:08.800 align:middle line:84% And so when we actually define measurable functions, 00:12:08.800 --> 00:12:15.310 align:middle line:84% we'll be using as input more of these half infinite intervals 00:12:15.310 --> 00:12:16.760 align:middle line:84% being measurable rather than this. 00:12:16.760 --> 00:12:18.550 align:middle line:90% So let's get to it. 00:12:18.550 --> 00:12:20.600 align:middle line:84% Let's define measurable function. 00:12:20.600 --> 00:12:29.950 align:middle line:84% So now just this comes with the territory. 00:12:29.950 --> 00:12:33.180 align:middle line:84% But we're going to be considering 00:12:33.180 --> 00:12:47.690 align:middle line:84% extended real valued functions in what we do. 00:12:47.690 --> 00:12:49.310 align:middle line:84% So I shouldn't have written that down. 00:12:49.310 --> 00:12:59.117 align:middle line:84% I should have written down extended real numbers. 00:12:59.117 --> 00:12:59.950 align:middle line:90% What does this mean? 00:12:59.950 --> 00:13:01.800 align:middle line:84% This just means the set of real numbers 00:13:01.800 --> 00:13:04.660 align:middle line:84% along with plus and minus infinity. 00:13:04.660 --> 00:13:08.130 align:middle line:84% So when I write the interval minus infinity, 00:13:08.130 --> 00:13:16.900 align:middle line:84% infinity, this is our union, the two symbols plus or minus 00:13:16.900 --> 00:13:18.540 align:middle line:90% infinity. 00:13:18.540 --> 00:13:21.380 align:middle line:84% Now we're going to have expressions 00:13:21.380 --> 00:13:24.710 align:middle line:84% where we allow a function to take on the value plus infinity 00:13:24.710 --> 00:13:26.760 align:middle line:90% or minus infinity. 00:13:26.760 --> 00:13:34.560 align:middle line:84% So we should set down what we mean when we multiply 00:13:34.560 --> 00:13:37.925 align:middle line:90% some of these things together. 00:13:37.925 --> 00:13:49.720 align:middle line:90% 00:13:49.720 --> 00:13:54.610 align:middle line:84% So sums are defined as x plus or minus 00:13:54.610 --> 00:13:59.050 align:middle line:84% infinity equals plus or minus infinity for all x in r. 00:13:59.050 --> 00:14:06.070 align:middle line:84% So if I have two functions whose values could 00:14:06.070 --> 00:14:09.520 align:middle line:84% be in the extended real numbers and I have one is finite 00:14:09.520 --> 00:14:11.920 align:middle line:84% and the other is infinite, then I just, 00:14:11.920 --> 00:14:17.350 align:middle line:84% by convention, their sum is defined to be plus infinity. 00:14:17.350 --> 00:14:21.650 align:middle line:90% 00:14:21.650 --> 00:14:24.530 align:middle line:84% But I'm not allowed to take infinity 00:14:24.530 --> 00:14:29.490 align:middle line:84% minus infinity or infinity plus minus infinity. 00:14:29.490 --> 00:14:47.030 align:middle line:84% So this and products are defined as we take the convention 00:14:47.030 --> 00:14:53.144 align:middle line:84% that zero times plus or minus infinity equals zero. 00:14:53.144 --> 00:14:55.810 align:middle line:90% 00:14:55.810 --> 00:15:00.460 align:middle line:84% And we won't come up against this 00:15:00.460 --> 00:15:02.950 align:middle line:84% until we discuss the integral, why 00:15:02.950 --> 00:15:04.450 align:middle line:90% we would need such expressions. 00:15:04.450 --> 00:15:08.110 align:middle line:84% But you could have a function that's, let's say, 00:15:08.110 --> 00:15:10.750 align:middle line:84% identically equal to infinity, identically gives 00:15:10.750 --> 00:15:12.490 align:middle line:90% this symbol plus infinity. 00:15:12.490 --> 00:15:16.000 align:middle line:84% If I multiply that function by zero, then I should get zero. 00:15:16.000 --> 00:15:19.270 align:middle line:84% This has nothing to do with limiting processes though. 00:15:19.270 --> 00:15:21.623 align:middle line:84% These are purely algebraic expressions 00:15:21.623 --> 00:15:22.540 align:middle line:90% we'll be dealing with. 00:15:22.540 --> 00:15:25.090 align:middle line:84% I'm not saying that everything you learned 00:15:25.090 --> 00:15:28.330 align:middle line:84% in 18100 Real Analysis about being careful when 00:15:28.330 --> 00:15:32.530 align:middle line:84% you have infinity over infinity or zero times infinity is-- 00:15:32.530 --> 00:15:34.810 align:middle line:90% I'm not saying throw that away. 00:15:34.810 --> 00:15:39.400 align:middle line:84% I'm just saying when we have certain algebraic expressions, 00:15:39.400 --> 00:15:42.490 align:middle line:84% these are the conventions that we adopt. 00:15:42.490 --> 00:15:47.280 align:middle line:84% And x times plus or minus infinity 00:15:47.280 --> 00:15:54.746 align:middle line:84% equals plus or minus infinity for all x in R take away zero. 00:15:54.746 --> 00:15:56.660 align:middle line:90% OK. 00:15:56.660 --> 00:16:01.135 align:middle line:84% And let me just recall for you what it means for-- 00:16:01.135 --> 00:16:07.190 align:middle line:90% 00:16:07.190 --> 00:16:09.830 align:middle line:84% we'll be making some limiting statements 00:16:09.830 --> 00:16:13.500 align:middle line:84% about certain numbers approaching other numbers. 00:16:13.500 --> 00:16:16.130 align:middle line:84% So what would it mean for a sequence of numbers 00:16:16.130 --> 00:16:19.850 align:middle line:84% to approach plus infinity or minus infinity? 00:16:19.850 --> 00:16:33.230 align:middle line:84% I want you to recall that a sequence a n of real numbers 00:16:33.230 --> 00:16:36.800 align:middle line:90% converges to infinity. 00:16:36.800 --> 00:16:40.250 align:middle line:84% And then you can make a similar definition for minus infinity 00:16:40.250 --> 00:16:50.440 align:middle line:84% if, for every R positive, there exists a natural number N such 00:16:50.440 --> 00:16:54.460 align:middle line:84% that for all N bigger than or equal to N 00:16:54.460 --> 00:16:58.914 align:middle line:90% a sub n is bigger than R. OK. 00:16:58.914 --> 00:17:03.190 align:middle line:90% 00:17:03.190 --> 00:17:03.690 align:middle line:90% OK. 00:17:03.690 --> 00:17:06.329 align:middle line:84% And I mean I've been using something equals infinity 00:17:06.329 --> 00:17:08.910 align:middle line:84% already when we've been just discussing 00:17:08.910 --> 00:17:10.710 align:middle line:84% measure, the outer measure of something 00:17:10.710 --> 00:17:12.750 align:middle line:90% being equal to infinity. 00:17:12.750 --> 00:17:14.700 align:middle line:84% In what follows, we'll be having expressions 00:17:14.700 --> 00:17:16.980 align:middle line:84% where we allow something to equal infinity. 00:17:16.980 --> 00:17:21.069 align:middle line:84% And we'll have algebraic expressions of this type 00:17:21.069 --> 00:17:23.160 align:middle line:84% so I just need to set down convention 00:17:23.160 --> 00:17:24.990 align:middle line:84% that if I have a real number plus or minus 00:17:24.990 --> 00:17:28.740 align:middle line:84% infinity, that is by definition equal to plus or minus 00:17:28.740 --> 00:17:29.310 align:middle line:90% infinity. 00:17:29.310 --> 00:17:31.980 align:middle line:84% If I have zero times something that 00:17:31.980 --> 00:17:35.490 align:middle line:84% equals plus or minus infinity, then that is by convention 00:17:35.490 --> 00:17:40.990 align:middle line:90% equal to zero and so on. 00:17:40.990 --> 00:17:41.590 align:middle line:90% All right. 00:17:41.590 --> 00:17:50.880 align:middle line:84% So measurable functions I said should 00:17:50.880 --> 00:17:54.900 align:middle line:84% be those functions at least motivated 00:17:54.900 --> 00:17:56.610 align:middle line:90% by this discussion before. 00:17:56.610 --> 00:18:00.690 align:middle line:84% Those functions or the functions we're interested in 00:18:00.690 --> 00:18:02.832 align:middle line:84% are those functions so that the inverse image 00:18:02.832 --> 00:18:04.290 align:middle line:84% of closed intervals are measurable. 00:18:04.290 --> 00:18:07.965 align:middle line:84% So that's how we'll define measurable functions almost. 00:18:07.965 --> 00:18:11.323 align:middle line:90% 00:18:11.323 --> 00:18:13.490 align:middle line:84% And the equivalent way, which is a little bit easier 00:18:13.490 --> 00:18:15.990 align:middle line:90% to work with, is the following. 00:18:15.990 --> 00:18:30.440 align:middle line:84% So let E be a measurable set and f from E to the extended reels. 00:18:30.440 --> 00:18:35.880 align:middle line:90% 00:18:35.880 --> 00:18:46.350 align:middle line:84% You say f is Lebesgue measurable-- 00:18:46.350 --> 00:18:54.960 align:middle line:84% that's new terminology-- if, for all alpha in R, 00:18:54.960 --> 00:19:02.360 align:middle line:84% if I take the inverse image of the half infinite interval, 00:19:02.360 --> 00:19:12.130 align:middle line:84% alpha to infinity, this is measurable I should say. 00:19:12.130 --> 00:19:15.010 align:middle line:84% We had some notation last time, the script 00:19:15.010 --> 00:19:19.315 align:middle line:84% M being the collection of measurable sets, i.e. 00:19:19.315 --> 00:19:22.100 align:middle line:90% 00:19:22.100 --> 00:19:24.320 align:middle line:90% is Lebesgue measurable. 00:19:24.320 --> 00:19:24.820 align:middle line:90% OK. 00:19:24.820 --> 00:19:29.210 align:middle line:90% 00:19:29.210 --> 00:19:31.930 align:middle line:84% As we'll see in a minute, this is an equivalent definition 00:19:31.930 --> 00:19:35.710 align:middle line:84% really of requiring that the inverse image 00:19:35.710 --> 00:19:42.340 align:middle line:84% of a closed and bounded interval is measurable, 00:19:42.340 --> 00:19:43.938 align:middle line:84% or at least we'll see one direction 00:19:43.938 --> 00:19:44.980 align:middle line:90% of why that's equivalent. 00:19:44.980 --> 00:19:48.250 align:middle line:84% And then in your own time, you can figure out 00:19:48.250 --> 00:19:50.620 align:middle line:90% why it's actually equivalent. 00:19:50.620 --> 00:19:58.480 align:middle line:90% So now why not include alpha? 00:19:58.480 --> 00:20:01.090 align:middle line:84% Why do I have to look at the inverse image of this half 00:20:01.090 --> 00:20:03.040 align:middle line:90% open interval? 00:20:03.040 --> 00:20:06.250 align:middle line:84% And why am I going from alpha to infinity and not, 00:20:06.250 --> 00:20:10.270 align:middle line:84% say, alpha to minus infinity or include alpha there? 00:20:10.270 --> 00:20:11.650 align:middle line:90% Why this specific kind? 00:20:11.650 --> 00:20:14.020 align:middle line:84% And then so what I want to first tell you 00:20:14.020 --> 00:20:20.180 align:middle line:84% is that looking at those types of sets 00:20:20.180 --> 00:20:21.740 align:middle line:84% and seeing if those are measurable 00:20:21.740 --> 00:20:24.270 align:middle line:84% is equivalent to this definition that I'm giving here. 00:20:24.270 --> 00:20:34.130 align:middle line:90% 00:20:34.130 --> 00:20:37.030 align:middle line:84% So let's take a function from a measurable set E, 00:20:37.030 --> 00:20:41.495 align:middle line:84% a subset of R to the extended reals. 00:20:41.495 --> 00:20:50.560 align:middle line:90% 00:20:50.560 --> 00:20:55.960 align:middle line:84% Then the following are equivalent. 00:20:55.960 --> 00:20:58.780 align:middle line:90% 00:20:58.780 --> 00:21:07.100 align:middle line:84% One is kind of the property that's 00:21:07.100 --> 00:21:08.750 align:middle line:84% in the definition of being measurable. 00:21:08.750 --> 00:21:19.460 align:middle line:84% For all alpha in R, f inverse of alpha infinity is measurable. 00:21:19.460 --> 00:21:20.480 align:middle line:90% Two. 00:21:20.480 --> 00:21:32.030 align:middle line:84% For all alpha in R, f inverse including alpha is measurable. 00:21:32.030 --> 00:21:34.810 align:middle line:90% 00:21:34.810 --> 00:21:36.280 align:middle line:90% Three. 00:21:36.280 --> 00:21:44.160 align:middle line:84% All alpha in R, the inverse image of minus 00:21:44.160 --> 00:21:48.000 align:middle line:90% infinity to alpha is measurable. 00:21:48.000 --> 00:21:54.310 align:middle line:90% 00:21:54.310 --> 00:22:01.880 align:middle line:84% And the last condition is, for all alpha in R, 00:22:01.880 --> 00:22:11.240 align:middle line:84% the inverse image of now include alpha is measurable. 00:22:11.240 --> 00:22:17.510 align:middle line:84% So to check what you get from this, 00:22:17.510 --> 00:22:19.740 align:middle line:84% to check that a function is Lebesgue measurable, 00:22:19.740 --> 00:22:26.380 align:middle line:84% it suffices not to just solely prove this property 00:22:26.380 --> 00:22:27.920 align:middle line:90% or check this property. 00:22:27.920 --> 00:22:30.460 align:middle line:84% You can check it on other types of sets. 00:22:30.460 --> 00:22:33.310 align:middle line:84% It suffices to only check it-- it suffices to check it 00:22:33.310 --> 00:22:36.370 align:middle line:84% on either these types of sets, or these types of sets, 00:22:36.370 --> 00:22:38.260 align:middle line:90% or these types of sets. 00:22:38.260 --> 00:22:44.650 align:middle line:84% So if you can prove that for all alpha, this type of set 00:22:44.650 --> 00:22:47.320 align:middle line:84% or the inverse image of these types of intervals 00:22:47.320 --> 00:22:50.050 align:middle line:84% are measurable, then it's equivalent to saying 00:22:50.050 --> 00:22:51.488 align:middle line:90% that f is Lebesgue measurable. 00:22:51.488 --> 00:22:51.988 align:middle line:90% OK. 00:22:51.988 --> 00:22:56.740 align:middle line:90% 00:22:56.740 --> 00:22:57.240 align:middle line:90% OK. 00:22:57.240 --> 00:23:00.610 align:middle line:90% 00:23:00.610 --> 00:23:02.710 align:middle line:84% Now the proof is not hard based on what 00:23:02.710 --> 00:23:05.500 align:middle line:84% we know about Lebesgue measurable sets 00:23:05.500 --> 00:23:08.480 align:middle line:84% which, again, we proved last time that they form a sigma 00:23:08.480 --> 00:23:08.980 align:middle line:90% algebra. 00:23:08.980 --> 00:23:11.120 align:middle line:84% They're closed under taking countable unions, 00:23:11.120 --> 00:23:13.670 align:middle line:84% intersections, and complements, and so on. 00:23:13.670 --> 00:23:19.980 align:middle line:84% So let's first prove that one implies two. 00:23:19.980 --> 00:23:22.500 align:middle line:84% So I'm saying they're all equivalent, which means they 00:23:22.500 --> 00:23:24.040 align:middle line:90% should all imply each other. 00:23:24.040 --> 00:23:28.130 align:middle line:84% So let's first prove one implies two. 00:23:28.130 --> 00:23:41.360 align:middle line:84% So suppose one holds, then for all alpha in R, 00:23:41.360 --> 00:23:44.090 align:middle line:84% now I want to check that two holds. 00:23:44.090 --> 00:23:50.630 align:middle line:84% I can write alpha to infinity as the intersection 00:23:50.630 --> 00:23:55.130 align:middle line:84% over natural numbers n of alpha minus 1 00:23:55.130 --> 00:24:00.320 align:middle line:84% over n infinity, which implies the great thing about taking 00:24:00.320 --> 00:24:10.190 align:middle line:84% inverse images is that they respect all operations you 00:24:10.190 --> 00:24:11.000 align:middle line:90% can do on set. 00:24:11.000 --> 00:24:13.790 align:middle line:84% So the inverse image of any type of intersection 00:24:13.790 --> 00:24:15.665 align:middle line:84% is the intersection of the inverse images. 00:24:15.665 --> 00:24:24.460 align:middle line:90% 00:24:24.460 --> 00:24:34.310 align:middle line:84% And if I'm assuming each of these is measurable by 1, 00:24:34.310 --> 00:24:37.070 align:middle line:84% then this is a countable intersection 00:24:37.070 --> 00:24:39.770 align:middle line:90% of Lebesgue measurable sets. 00:24:39.770 --> 00:24:42.770 align:middle line:84% Again, when I say what is the Lebesgue measurable sets, 00:24:42.770 --> 00:24:46.040 align:middle line:84% I'm talking about the inverse image of this set, not 00:24:46.040 --> 00:24:47.630 align:middle line:90% this set in here. 00:24:47.630 --> 00:24:50.790 align:middle line:84% I'm talking about the inverse image of this interval. 00:24:50.790 --> 00:24:53.120 align:middle line:84% So each of those is a Lebesgue measurable 00:24:53.120 --> 00:24:58.260 align:middle line:84% set so their intersection is Lebesgue measurable. 00:24:58.260 --> 00:25:00.210 align:middle line:90% OK. 00:25:00.210 --> 00:25:02.430 align:middle line:84% And therefore for every alpha in R, 00:25:02.430 --> 00:25:05.340 align:middle line:84% the inverse image of these closed intervals 00:25:05.340 --> 00:25:09.660 align:middle line:90% are measurable. 00:25:09.660 --> 00:25:13.380 align:middle line:84% And how do I show two implies one? 00:25:13.380 --> 00:25:15.675 align:middle line:90% By a similar game. 00:25:15.675 --> 00:25:18.970 align:middle line:90% 00:25:18.970 --> 00:25:19.870 align:middle line:90% Suppose two holds. 00:25:19.870 --> 00:25:22.470 align:middle line:90% 00:25:22.470 --> 00:25:31.780 align:middle line:84% Then for all alpha in R, the half open interval now 00:25:31.780 --> 00:25:38.070 align:middle line:84% I can write as the union over n alpha plus 1 over n 00:25:38.070 --> 00:25:39.000 align:middle line:90% to infinity. 00:25:39.000 --> 00:25:42.710 align:middle line:90% 00:25:42.710 --> 00:25:48.220 align:middle line:84% And therefore the inverse image of alpha to infinity 00:25:48.220 --> 00:25:52.420 align:middle line:84% is equal to the union of the inverse images of these sets. 00:25:52.420 --> 00:25:58.800 align:middle line:90% 00:25:58.800 --> 00:26:03.000 align:middle line:84% And again, if I'm assuming two, then I'm assuming each of these 00:26:03.000 --> 00:26:06.090 align:middle line:84% for each n is a Lebesgue measurable set. 00:26:06.090 --> 00:26:09.210 align:middle line:84% And we know countable unions of Lebesgue measurable sets 00:26:09.210 --> 00:26:10.620 align:middle line:90% are again Lebesgue measurable. 00:26:10.620 --> 00:26:14.760 align:middle line:90% So this is Lebesgue measurable. 00:26:14.760 --> 00:26:15.390 align:middle line:90% OK. 00:26:15.390 --> 00:26:20.540 align:middle line:84% So that proves one and two are equivalent. 00:26:20.540 --> 00:26:31.610 align:middle line:90% 00:26:31.610 --> 00:26:33.290 align:middle line:90% OK. 00:26:33.290 --> 00:26:39.520 align:middle line:90% And we get three and four. 00:26:39.520 --> 00:26:46.170 align:middle line:84% So first off, two is equivalent to three 00:26:46.170 --> 00:26:50.850 align:middle line:90% simply because if I take-- 00:26:50.850 --> 00:26:55.750 align:middle line:84% let's see-- minus infinity to alpha, 00:26:55.750 --> 00:27:04.710 align:middle line:84% this is equal to alpha infinity complement. 00:27:04.710 --> 00:27:06.900 align:middle line:84% And therefore again, when I take inverse images, 00:27:06.900 --> 00:27:10.140 align:middle line:84% and knowing that that the complement of a Lebesgue 00:27:10.140 --> 00:27:12.690 align:middle line:84% measurable set is Lebesgue measurable, 00:27:12.690 --> 00:27:20.180 align:middle line:84% I get that two is equivalent to three and one 00:27:20.180 --> 00:27:23.080 align:middle line:90% is equivalent to four. 00:27:23.080 --> 00:27:27.820 align:middle line:84% Again, since I have this last type of set, 00:27:27.820 --> 00:27:31.510 align:middle line:84% minus infinity to alpha, this is equal to-- so this 00:27:31.510 --> 00:27:37.940 align:middle line:84% is for all alpha in R since for all alpha in R, 00:27:37.940 --> 00:27:43.080 align:middle line:90% this is equal to complement. 00:27:43.080 --> 00:27:47.300 align:middle line:84% So again, if I take the inverse image of this set, 00:27:47.300 --> 00:27:50.990 align:middle line:84% I will get the complement of the inverse image of this set. 00:27:50.990 --> 00:27:53.983 align:middle line:84% And if that's measurable, that inverse image is measurable, 00:27:53.983 --> 00:27:55.400 align:middle line:84% then its complement is measurable, 00:27:55.400 --> 00:27:57.680 align:middle line:90% implying that's measurable. 00:27:57.680 --> 00:28:02.510 align:middle line:84% So it's a simple game or it's a simple proof just based 00:28:02.510 --> 00:28:06.170 align:middle line:84% on the fact that we know that Lebesgue measurable 00:28:06.170 --> 00:28:08.430 align:middle line:84% sets are closed undertaking countable unions, 00:28:08.430 --> 00:28:10.955 align:middle line:90% intersections, and complements. 00:28:10.955 --> 00:28:11.455 align:middle line:90% OK. 00:28:11.455 --> 00:28:18.420 align:middle line:90% 00:28:18.420 --> 00:28:21.270 align:middle line:84% So then we get the following theorem 00:28:21.270 --> 00:28:29.730 align:middle line:84% that, if E is measurable, so E is a subset of R is measurable, 00:28:29.730 --> 00:28:39.030 align:middle line:84% and f from E to R is a measurable function-- 00:28:39.030 --> 00:28:46.055 align:middle line:90% 00:28:46.055 --> 00:28:47.680 align:middle line:84% in the future, I will probably just say 00:28:47.680 --> 00:28:50.650 align:middle line:84% f is measurable, not a measurable function-- 00:28:50.650 --> 00:28:55.410 align:middle line:90% 00:28:55.410 --> 00:29:12.930 align:middle line:84% then for all f in the Borel sigma algebra, f inverse of F 00:29:12.930 --> 00:29:13.680 align:middle line:90% is measurable. 00:29:13.680 --> 00:29:18.120 align:middle line:84% The inverse image of f is measurable. 00:29:18.120 --> 00:29:18.840 align:middle line:90% OK. 00:29:18.840 --> 00:29:23.275 align:middle line:90% So what's the proof? 00:29:23.275 --> 00:29:33.120 align:middle line:90% 00:29:33.120 --> 00:29:41.560 align:middle line:84% f measurable implies that, for all ab, a less 00:29:41.560 --> 00:29:49.040 align:middle line:84% b, f inverse of the open interval ab, which 00:29:49.040 --> 00:30:06.910 align:middle line:84% is equal to the f inverse of the two intersect equals 00:30:06.910 --> 00:30:08.050 align:middle line:90% f inverse of-- 00:30:08.050 --> 00:30:18.610 align:middle line:90% 00:30:18.610 --> 00:30:23.050 align:middle line:84% So if I take the inverse image of this open interval ab, 00:30:23.050 --> 00:30:26.282 align:middle line:84% then it's equal to the inverse image of this intersection, 00:30:26.282 --> 00:30:28.240 align:middle line:84% which is equal to the intersection of these two 00:30:28.240 --> 00:30:29.360 align:middle line:90% inverse images. 00:30:29.360 --> 00:30:31.700 align:middle line:84% And if I'm assuming f is measurable, 00:30:31.700 --> 00:30:34.150 align:middle line:90% then each of these sets-- 00:30:34.150 --> 00:30:37.600 align:middle line:84% each of these pre-images are measurable 00:30:37.600 --> 00:30:43.920 align:middle line:84% by the previous theorem that I just proved 00:30:43.920 --> 00:30:48.730 align:middle line:90% and therefore that's measurable. 00:30:48.730 --> 00:31:01.500 align:middle line:84% So I've shown that open intervals are measurable. 00:31:01.500 --> 00:31:08.650 align:middle line:90% 00:31:08.650 --> 00:31:15.820 align:middle line:84% And similar to how we concluded that open sets are measurable, 00:31:15.820 --> 00:31:17.140 align:middle line:90% you can then-- 00:31:17.140 --> 00:31:19.780 align:middle line:84% we use the fact that every open set, which 00:31:19.780 --> 00:31:23.140 align:middle line:84% you proved in assignment 3, that every open set can be written 00:31:23.140 --> 00:31:31.790 align:middle line:84% as a countable union of disjoint open intervals, 00:31:31.790 --> 00:31:43.650 align:middle line:84% thus f inverse of u is measurable for all open sets, 00:31:43.650 --> 00:31:47.920 align:middle line:90% subsets of R. OK. 00:31:47.920 --> 00:31:50.440 align:middle line:90% 00:31:50.440 --> 00:31:56.180 align:middle line:90% And since the set-- 00:31:56.180 --> 00:32:02.390 align:middle line:84% let's call it A, which is the set of all F such 00:32:02.390 --> 00:32:09.860 align:middle line:90% that this is a-- 00:32:09.860 --> 00:32:12.350 align:middle line:84% so you're also proving in the assignment 00:32:12.350 --> 00:32:16.670 align:middle line:84% this collection of sets, such that the inverse images are 00:32:16.670 --> 00:32:30.610 align:middle line:84% Lebesgue measurable, this is a sigma algebra containing-- 00:32:30.610 --> 00:32:37.780 align:middle line:84% since we've now proved that all open sets are in the set, 00:32:37.780 --> 00:32:43.040 align:middle line:84% this implies that the Borel sigma algebra, which is, again, 00:32:43.040 --> 00:32:46.760 align:middle line:84% the smallest sigma algebra containing all open sets, 00:32:46.760 --> 00:32:52.550 align:middle line:84% is a subset of this here, which is 00:32:52.550 --> 00:32:54.268 align:middle line:90% the statement of the theorem. 00:32:54.268 --> 00:32:54.768 align:middle line:90% OK. 00:32:54.768 --> 00:33:00.210 align:middle line:90% 00:33:00.210 --> 00:33:01.550 align:middle line:90% All right. 00:33:01.550 --> 00:33:10.590 align:middle line:84% And so let me just add on here another thing that-- 00:33:10.590 --> 00:33:11.090 align:middle line:90% OK. 00:33:11.090 --> 00:33:15.110 align:middle line:84% So now we know that if we have a measurable function, 00:33:15.110 --> 00:33:21.460 align:middle line:84% it takes the inverse image of Borel sets. 00:33:21.460 --> 00:33:24.760 align:middle line:84% So sets that belong in the Borel sigma algebra are measurable. 00:33:24.760 --> 00:33:26.155 align:middle line:90% This includes all open sets. 00:33:26.155 --> 00:33:28.870 align:middle line:90% 00:33:28.870 --> 00:33:34.220 align:middle line:84% What about throwing plus or minus infinity into the mix? 00:33:34.220 --> 00:33:47.120 align:middle line:84% So if f goes from one measurable set E to R is measurable, 00:33:47.120 --> 00:33:59.110 align:middle line:90% then the sets-- 00:33:59.110 --> 00:34:00.470 align:middle line:90% well, let me write it this way. 00:34:00.470 --> 00:34:11.670 align:middle line:84% The inverse image of where f is infinite or equal 00:34:11.670 --> 00:34:16.020 align:middle line:84% to minus infinity, these are measurable as well. 00:34:16.020 --> 00:34:19.429 align:middle line:90% And how does one prove that? 00:34:19.429 --> 00:34:20.134 align:middle line:90% So proof. 00:34:20.134 --> 00:34:24.530 align:middle line:90% 00:34:24.530 --> 00:34:27.380 align:middle line:84% Again, we could just use the definition of how this works 00:34:27.380 --> 00:34:30.409 align:middle line:84% and the fact that we know Lebesgue measurable functions, 00:34:30.409 --> 00:34:33.530 align:middle line:84% or Lebesgue measurable sets, are closed 00:34:33.530 --> 00:34:36.469 align:middle line:84% undertaking countable intersections, unions, 00:34:36.469 --> 00:34:37.520 align:middle line:90% and complements. 00:34:37.520 --> 00:34:42.909 align:middle line:90% 00:34:42.909 --> 00:34:49.050 align:middle line:84% We have that f inverse of infinity, 00:34:49.050 --> 00:34:52.500 align:middle line:84% this is equal to the intersection overall n 00:34:52.500 --> 00:35:01.270 align:middle line:84% of f inverse, the inverse image of n to infinity. 00:35:01.270 --> 00:35:03.780 align:middle line:90% 00:35:03.780 --> 00:35:08.120 align:middle line:84% And if I'm assuming f is measurable, then each of these 00:35:08.120 --> 00:35:09.920 align:middle line:90% is a measurable set. 00:35:09.920 --> 00:35:13.010 align:middle line:84% The inverse image of n to infinity is a measurable set. 00:35:13.010 --> 00:35:17.580 align:middle line:84% I will often use inverse image or pre-image. 00:35:17.580 --> 00:35:23.610 align:middle line:84% These are meant to be taken as saying the same thing. 00:35:23.610 --> 00:35:25.140 align:middle line:84% Each of these is a measurable set 00:35:25.140 --> 00:35:28.920 align:middle line:84% so it's the countable intersection of measurable sets 00:35:28.920 --> 00:35:30.670 align:middle line:90% is measurable. 00:35:30.670 --> 00:35:34.880 align:middle line:90% So that's measurable. 00:35:34.880 --> 00:35:39.950 align:middle line:84% And similarly, if I look at the set 00:35:39.950 --> 00:35:41.690 align:middle line:84% where it's equal to minus infinity, 00:35:41.690 --> 00:35:43.730 align:middle line:84% this is equal to the intersection 00:35:43.730 --> 00:35:47.270 align:middle line:84% of the inverse image of minus infinity to n. 00:35:47.270 --> 00:35:52.250 align:middle line:90% 00:35:52.250 --> 00:35:56.030 align:middle line:84% Let's take these positives, so then I'll put minus n here. 00:35:56.030 --> 00:35:59.960 align:middle line:84% And by this theorem that we just proved a minute ago, 00:35:59.960 --> 00:36:02.040 align:middle line:90% each of these-- 00:36:02.040 --> 00:36:07.460 align:middle line:84% well, I mean we just proved that the inverse image of Borel sets 00:36:07.460 --> 00:36:10.370 align:middle line:90% are Lebesgue measurable so-- 00:36:10.370 --> 00:36:12.030 align:middle line:90% well, that won't apply. 00:36:12.030 --> 00:36:12.530 align:middle line:90% Never mind. 00:36:12.530 --> 00:36:13.790 align:middle line:90% Forget what I just said. 00:36:13.790 --> 00:36:15.440 align:middle line:84% The theorem we just proved a minute ago 00:36:15.440 --> 00:36:19.490 align:middle line:84% that Lebesgue measurable functions take 00:36:19.490 --> 00:36:24.140 align:middle line:84% sets of this type to Lebesgue measurable sets 00:36:24.140 --> 00:36:27.950 align:middle line:84% implies that each of these is Lebesgue measurable 00:36:27.950 --> 00:36:32.820 align:middle line:84% and therefore it's a union of Lebesgue measurable sets, 00:36:32.820 --> 00:36:38.360 align:middle line:84% so for all n, so that's measurable. 00:36:38.360 --> 00:36:43.060 align:middle line:90% 00:36:43.060 --> 00:36:45.670 align:middle line:84% So if I have an extended real value 00:36:45.670 --> 00:36:52.770 align:middle line:84% function, and the inverse image of every Borel set-- 00:36:52.770 --> 00:36:57.800 align:middle line:84% you can even toss in the two infinities if you like. 00:36:57.800 --> 00:37:01.190 align:middle line:84% The inverse image of those sets is Lebesgue measurable. 00:37:01.190 --> 00:37:04.820 align:middle line:84% So in particular, we conclude that the inverse images 00:37:04.820 --> 00:37:09.680 align:middle line:84% of closed and bounded intervals is Lebesgue measurable 00:37:09.680 --> 00:37:12.560 align:middle line:90% for measurable functions. 00:37:12.560 --> 00:37:13.060 align:middle line:90% OK. 00:37:13.060 --> 00:37:16.270 align:middle line:90% Now what are the simplest types? 00:37:16.270 --> 00:37:17.990 align:middle line:90% Again, you see a definition. 00:37:17.990 --> 00:37:19.730 align:middle line:90% You should ask for an example. 00:37:19.730 --> 00:37:23.920 align:middle line:84% What are the simplest types of measurable functions? 00:37:23.920 --> 00:37:37.560 align:middle line:84% So if f from R to R is continuous, 00:37:37.560 --> 00:37:43.270 align:middle line:84% then that implies that f is measurable. 00:37:43.270 --> 00:37:46.760 align:middle line:84% So in the end when we build our definition of the Lebesgue 00:37:46.760 --> 00:37:52.110 align:middle line:84% integral, we should encompass Reimann integration as well. 00:37:52.110 --> 00:37:55.670 align:middle line:84% In other words, we should be able to also integrate 00:37:55.670 --> 00:37:57.500 align:middle line:90% continuous functions. 00:37:57.500 --> 00:38:00.620 align:middle line:84% And later we'll see, and something that we want as well, 00:38:00.620 --> 00:38:03.260 align:middle line:84% is that the Lebesgue integral of a continuous function 00:38:03.260 --> 00:38:05.990 align:middle line:84% should reduce down to the Reimann integral 00:38:05.990 --> 00:38:08.940 align:middle line:90% of a continuous function. 00:38:08.940 --> 00:38:12.770 align:middle line:84% So at a minimum, when we're building these concepts, 00:38:12.770 --> 00:38:15.260 align:middle line:84% we should check as a kind of sanity check 00:38:15.260 --> 00:38:17.840 align:middle line:84% that we are including continuous functions 00:38:17.840 --> 00:38:22.350 align:middle line:84% in these possible functions which we will integrate. 00:38:22.350 --> 00:38:26.610 align:middle line:84% So if f is continuous, f is measurable, why is this? 00:38:26.610 --> 00:38:29.120 align:middle line:84% This is because for all alpha in R, 00:38:29.120 --> 00:38:36.390 align:middle line:84% if I look at the inverse image of alpha infinity-- 00:38:36.390 --> 00:38:42.510 align:middle line:84% so first off, this is equal to this set. 00:38:42.510 --> 00:38:49.750 align:middle line:84% And this is an open set so this is open. 00:38:49.750 --> 00:38:54.130 align:middle line:90% 00:38:54.130 --> 00:38:57.910 align:middle line:84% So the inverse image of an open set for a continuous function 00:38:57.910 --> 00:38:58.990 align:middle line:90% is open. 00:38:58.990 --> 00:39:03.165 align:middle line:90% And therefore it's measurable. 00:39:03.165 --> 00:39:03.665 align:middle line:90% Right? 00:39:03.665 --> 00:39:08.590 align:middle line:90% 00:39:08.590 --> 00:39:09.090 align:middle line:90% OK. 00:39:09.090 --> 00:39:16.180 align:middle line:90% 00:39:16.180 --> 00:39:18.880 align:middle line:84% So how about a different example? 00:39:18.880 --> 00:39:27.990 align:middle line:90% 00:39:27.990 --> 00:39:32.640 align:middle line:84% Let's take a measurable subset E of R. 00:39:32.640 --> 00:39:39.640 align:middle line:84% Let F be another measurable subset. 00:39:39.640 --> 00:39:47.660 align:middle line:90% 00:39:47.660 --> 00:39:51.410 align:middle line:84% And define the indicator function 00:39:51.410 --> 00:40:07.530 align:middle line:84% chi of F of x to be 1 if x is in F, 0 if x is not in F. Then 00:40:07.530 --> 00:40:08.850 align:middle line:90% this function chi F-- 00:40:08.850 --> 00:40:11.890 align:middle line:90% 00:40:11.890 --> 00:40:21.240 align:middle line:84% now if I think of it as being a function from E to R, 00:40:21.240 --> 00:40:24.740 align:middle line:90% this is measurable. 00:40:24.740 --> 00:40:25.607 align:middle line:90% Now, why is that? 00:40:25.607 --> 00:40:26.690 align:middle line:90% Well, we can just compute. 00:40:26.690 --> 00:40:31.400 align:middle line:90% 00:40:31.400 --> 00:40:38.150 align:middle line:84% If alpha is in R, if I look at the inverse image of alpha 00:40:38.150 --> 00:40:43.370 align:middle line:84% to infinity, this is equal to one of three things. 00:40:43.370 --> 00:40:47.930 align:middle line:84% Since chi takes on only the values 1 and 0, if alpha 00:40:47.930 --> 00:40:50.550 align:middle line:84% is bigger than 1, this is the empty set, 00:40:50.550 --> 00:40:56.970 align:middle line:84% which is measurable if alpha's bigger than or equal to 1 00:40:56.970 --> 00:40:58.770 align:middle line:84% since this set does not include-- 00:40:58.770 --> 00:41:01.480 align:middle line:90% would not include 1. 00:41:01.480 --> 00:41:07.400 align:middle line:84% If alpha is between 0 and less than 1, 00:41:07.400 --> 00:41:11.580 align:middle line:84% then the inverse image of this set is equal to E intersect 00:41:11.580 --> 00:41:13.820 align:middle line:90% F. E is measurable. 00:41:13.820 --> 00:41:14.820 align:middle line:90% F is measurable. 00:41:14.820 --> 00:41:17.360 align:middle line:84% So their intersection is measurable. 00:41:17.360 --> 00:41:24.230 align:middle line:84% And if alpha's less than 0, then the inverse image of this set-- 00:41:24.230 --> 00:41:27.640 align:middle line:84% in other words, what sets map into-- 00:41:27.640 --> 00:41:30.190 align:middle line:84% from one negative number to infinity-- well, 00:41:30.190 --> 00:41:31.910 align:middle line:90% both 1 and 0 map into there. 00:41:31.910 --> 00:41:42.370 align:middle line:84% And that's the entire set E. And so whatever you-- 00:41:42.370 --> 00:41:47.470 align:middle line:84% no matter what alpha is, the inverse image of these-- 00:41:47.470 --> 00:41:49.090 align:middle line:90% of that set is measurable. 00:41:49.090 --> 00:42:02.150 align:middle line:90% 00:42:02.150 --> 00:42:05.720 align:middle line:84% So these are basic properties of measurable functions. 00:42:05.720 --> 00:42:13.590 align:middle line:84% Let's continue what would we like more-- 00:42:13.590 --> 00:42:15.930 align:middle line:84% so what other properties of measurable functions, again, 00:42:15.930 --> 00:42:18.300 align:middle line:84% which we would hope to integrate or we 00:42:18.300 --> 00:42:22.200 align:middle line:84% will be integrating at least a class of these in the end 00:42:22.200 --> 00:42:23.070 align:middle line:90% to satisfy. 00:42:23.070 --> 00:42:25.830 align:middle line:84% Well, we would like them to be closed, 00:42:25.830 --> 00:42:32.300 align:middle line:84% undertaking linear combinations, and also products, 00:42:32.300 --> 00:42:34.520 align:middle line:84% because in the end, we'll have LP spaces, which 00:42:34.520 --> 00:42:37.460 align:middle line:90% is products of-- 00:42:37.460 --> 00:42:40.640 align:middle line:84% integrals of products of integral functions. 00:42:40.640 --> 00:42:43.520 align:middle line:90% 00:42:43.520 --> 00:42:44.920 align:middle line:84% So we have the following theorem. 00:42:44.920 --> 00:42:53.470 align:middle line:90% 00:42:53.470 --> 00:43:00.020 align:middle line:84% So let's suppose E is measurable. 00:43:00.020 --> 00:43:02.360 align:middle line:84% And I have two functions going from E 00:43:02.360 --> 00:43:06.640 align:middle line:90% to R that are measurable. 00:43:06.640 --> 00:43:11.190 align:middle line:90% 00:43:11.190 --> 00:43:25.470 align:middle line:84% And I have a scalar in R. Then c times f, f 00:43:25.470 --> 00:43:31.480 align:middle line:90% plus g, and f times g-- 00:43:31.480 --> 00:43:33.550 align:middle line:84% so these are now all functions from E to R-- 00:43:33.550 --> 00:43:36.830 align:middle line:90% 00:43:36.830 --> 00:43:40.100 align:middle line:84% are measurable, are measurable functions. 00:43:40.100 --> 00:43:43.010 align:middle line:90% 00:43:43.010 --> 00:43:46.040 align:middle line:90% So what's the proof? 00:43:46.040 --> 00:43:51.170 align:middle line:84% Let's start with scalar multiplication. 00:43:51.170 --> 00:43:53.240 align:middle line:90% This equals 0. 00:43:53.240 --> 00:43:57.958 align:middle line:84% And c times f equals 0, which is a continuous function. 00:43:57.958 --> 00:43:59.000 align:middle line:90% It's a constant function. 00:43:59.000 --> 00:44:11.460 align:middle line:84% 0 is continuous, hence measurable. 00:44:11.460 --> 00:44:13.300 align:middle line:90% So let's suppose c is non-zero. 00:44:13.300 --> 00:44:15.520 align:middle line:84% So the-- we're out of the silly case. 00:44:15.520 --> 00:44:20.730 align:middle line:90% 00:44:20.730 --> 00:44:26.250 align:middle line:84% If c does not equal 0 and alpha is 00:44:26.250 --> 00:44:33.360 align:middle line:84% an element of R, then c times f of x greater than alpha-- 00:44:33.360 --> 00:44:39.840 align:middle line:84% this is equivalent to f of x is greater than alpha over c. 00:44:39.840 --> 00:44:47.380 align:middle line:84% So this implies that if I want the inverse image, if I look 00:44:47.380 --> 00:44:51.370 align:middle line:84% at c times f and I look at the inverse image of alpha 00:44:51.370 --> 00:44:59.310 align:middle line:84% to infinity, this is equal to the inverse image of-- by f 00:44:59.310 --> 00:45:04.560 align:middle line:84% of the set alpha over c infinity. 00:45:04.560 --> 00:45:08.860 align:middle line:90% 00:45:08.860 --> 00:45:11.300 align:middle line:84% And now if I'm assuming f is measurable, 00:45:11.300 --> 00:45:14.900 align:middle line:84% then the inverse image of this set is measurable. 00:45:14.900 --> 00:45:17.800 align:middle line:90% So that's a measurable set. 00:45:17.800 --> 00:45:22.010 align:middle line:84% And therefore, c times f is a measurable function. 00:45:22.010 --> 00:45:25.270 align:middle line:90% 00:45:25.270 --> 00:45:29.220 align:middle line:84% So now, let's look at the function f plus g. 00:45:29.220 --> 00:45:34.260 align:middle line:90% 00:45:34.260 --> 00:45:41.540 align:middle line:84% Suppose alpha's in R. Then let's do something similar. 00:45:41.540 --> 00:45:51.380 align:middle line:84% Then f of x plus g of x is greater than alpha. 00:45:51.380 --> 00:45:57.050 align:middle line:84% This is if and only if f of x is greater than alpha 00:45:57.050 --> 00:46:00.520 align:middle line:90% minus g of x. 00:46:00.520 --> 00:46:02.600 align:middle line:90% So I didn't say this. 00:46:02.600 --> 00:46:04.780 align:middle line:84% But when I'm looking at such a condition, 00:46:04.780 --> 00:46:08.450 align:middle line:84% I'm looking at those x's that will be in this inverse image. 00:46:08.450 --> 00:46:10.930 align:middle line:84% So I'm just trying to figure out an equivalent 00:46:10.930 --> 00:46:13.570 align:middle line:84% way of expressing this condition, which 00:46:13.570 --> 00:46:14.870 align:middle line:90% we saw was this condition. 00:46:14.870 --> 00:46:16.960 align:middle line:84% So what I'm really looking at is those 00:46:16.960 --> 00:46:20.630 align:middle line:84% x's which would lie in the-- this pre-image here. 00:46:20.630 --> 00:46:22.420 align:middle line:84% And what I did a minute ago was show it's 00:46:22.420 --> 00:46:25.010 align:middle line:90% equal to this pre-image here. 00:46:25.010 --> 00:46:29.410 align:middle line:84% So this is why I'm considering f of x plus g of x plus alpha. 00:46:29.410 --> 00:46:30.790 align:middle line:90% This is the condition that-- 00:46:30.790 --> 00:46:34.000 align:middle line:90% 00:46:34.000 --> 00:46:42.390 align:middle line:84% so maybe I'll just write this out. 00:46:42.390 --> 00:46:46.010 align:middle line:84% Then x is in this set if and only if f of x plus g of x 00:46:46.010 --> 00:46:49.280 align:middle line:84% is bigger than alpha, which is equivalent to f of x plus-- 00:46:49.280 --> 00:46:52.280 align:middle line:84% f of x is greater than alpha minus g of x. 00:46:52.280 --> 00:46:59.680 align:middle line:90% Now, you learned in 18.100-- 00:46:59.680 --> 00:47:03.550 align:middle line:90% A, B, P, Q, whatever-- 00:47:03.550 --> 00:47:07.778 align:middle line:84% that if I have any two real numbers, one bigger 00:47:07.778 --> 00:47:09.820 align:middle line:84% than the other, then I can find a rational number 00:47:09.820 --> 00:47:11.650 align:middle line:90% in between them. 00:47:11.650 --> 00:47:15.650 align:middle line:84% So if this number is bigger than this number, 00:47:15.650 --> 00:47:21.830 align:middle line:84% there exists a rational number, r, such that f of x 00:47:21.830 --> 00:47:28.100 align:middle line:84% is greater than r is greater than alpha minus g of x. 00:47:28.100 --> 00:47:31.100 align:middle line:84% So this-- assuming this does imply this. 00:47:31.100 --> 00:47:34.550 align:middle line:84% And of course, this condition also implies this condition. 00:47:34.550 --> 00:47:36.950 align:middle line:84% If there exists a rational number so that f of x 00:47:36.950 --> 00:47:39.260 align:middle line:84% is bigger than r is bigger than alpha minus g of x, 00:47:39.260 --> 00:47:42.380 align:middle line:84% then, of course, f of x is bigger than alpha minus g of x. 00:47:42.380 --> 00:47:44.720 align:middle line:84% So these two conditions are equivalent. 00:47:44.720 --> 00:47:48.450 align:middle line:90% 00:47:48.450 --> 00:47:56.110 align:middle line:84% So this is equivalent to saying there exists an r in Q such 00:47:56.110 --> 00:48:08.090 align:middle line:84% that x is in the inverse image of f r to infinity intersect 00:48:08.090 --> 00:48:14.400 align:middle line:84% and r is bigger than alpha minus g of x, which one can 00:48:14.400 --> 00:48:20.520 align:middle line:84% state as x is in the inverse image of alpha minus r 00:48:20.520 --> 00:48:21.540 align:middle line:90% to infinity. 00:48:21.540 --> 00:48:28.980 align:middle line:90% 00:48:28.980 --> 00:48:40.020 align:middle line:84% So this last expression here-- so let me come over here 00:48:40.020 --> 00:48:46.235 align:middle line:90% and started erasing. 00:48:46.235 --> 00:48:51.670 align:middle line:90% 00:48:51.670 --> 00:48:58.430 align:middle line:84% So we've shown that x is in the inverse image of this set, f 00:48:58.430 --> 00:49:01.360 align:middle line:84% plus g, the inverse image by f plus g of this set, 00:49:01.360 --> 00:49:04.510 align:middle line:84% if and only if there exists a rational number so that x 00:49:04.510 --> 00:49:06.610 align:middle line:84% is in the intersection of these two types of sets, 00:49:06.610 --> 00:49:08.920 align:middle line:90% which we know are measurable. 00:49:08.920 --> 00:49:14.420 align:middle line:84% So we can express this, or summarizing 00:49:14.420 --> 00:49:17.510 align:middle line:90% what we just found-- 00:49:17.510 --> 00:49:26.090 align:middle line:84% that f plus g inverse image of alpha infinity-- 00:49:26.090 --> 00:49:29.840 align:middle line:84% this is equal to the union over rational numbers 00:49:29.840 --> 00:49:33.200 align:middle line:90% Q, r and Q, such that-- 00:49:33.200 --> 00:49:36.680 align:middle line:84% or the inverse image of f inverse or the inverse image 00:49:36.680 --> 00:49:51.980 align:middle line:84% of r infinity intersect alpha minus r infinity-- 00:49:51.980 --> 00:49:55.290 align:middle line:90% 00:49:55.290 --> 00:49:58.020 align:middle line:84% so that's just, again, expressing 00:49:58.020 --> 00:50:00.260 align:middle line:90% what we did over there. 00:50:00.260 --> 00:50:02.180 align:middle line:90% And now what do we know? 00:50:02.180 --> 00:50:04.350 align:middle line:84% If we're assuming f is measurable, 00:50:04.350 --> 00:50:07.660 align:middle line:84% then this whole set is measurable. 00:50:07.660 --> 00:50:09.520 align:middle line:84% And we're assuming g is measurable. 00:50:09.520 --> 00:50:11.890 align:middle line:90% So this whole set is measurable. 00:50:11.890 --> 00:50:15.670 align:middle line:84% And therefore, this intersection of these two sets 00:50:15.670 --> 00:50:17.020 align:middle line:90% is measurable. 00:50:17.020 --> 00:50:19.282 align:middle line:84% And now I have countable union because, again, 00:50:19.282 --> 00:50:20.740 align:middle line:84% the rational numbers are countable. 00:50:20.740 --> 00:50:23.620 align:middle line:84% One of the first things you prove in analysis 00:50:23.620 --> 00:50:27.430 align:middle line:84% is that the set of rational numbers is countable. 00:50:27.430 --> 00:50:30.910 align:middle line:84% This is a countable union of measurable sets. 00:50:30.910 --> 00:50:32.110 align:middle line:90% So that's measurable. 00:50:32.110 --> 00:50:49.600 align:middle line:90% 00:50:49.600 --> 00:50:50.950 align:middle line:90% Now, what about f times g? 00:50:50.950 --> 00:50:53.095 align:middle line:90% Here, we'll pull a little trick. 00:50:53.095 --> 00:50:58.530 align:middle line:90% 00:50:58.530 --> 00:51:01.870 align:middle line:84% So we'll prove that if f is measurable, 00:51:01.870 --> 00:51:03.120 align:middle line:90% then its square is measurable. 00:51:03.120 --> 00:51:06.000 align:middle line:84% And then we'll use a simple identity 00:51:06.000 --> 00:51:08.760 align:middle line:84% to conclude that f times g is measurable. 00:51:08.760 --> 00:51:12.160 align:middle line:90% 00:51:12.160 --> 00:51:19.400 align:middle line:84% So now I claim that f squared is measurable. 00:51:19.400 --> 00:51:22.590 align:middle line:90% 00:51:22.590 --> 00:51:26.460 align:middle line:84% Let alpha be a subset of R. If alpha 00:51:26.460 --> 00:51:32.590 align:middle line:84% is bigger than or equal to 0, well, let's do this stupid case 00:51:32.590 --> 00:51:33.090 align:middle line:90% first. 00:51:33.090 --> 00:51:40.980 align:middle line:84% If alpha is less than 0, then f squared-- 00:51:40.980 --> 00:51:43.780 align:middle line:90% this is a non-negative function. 00:51:43.780 --> 00:51:48.730 align:middle line:84% So if I take the inverse image of alpha to infinity, 00:51:48.730 --> 00:51:55.740 align:middle line:84% this is just equal to the domain E. So this is measurable. 00:51:55.740 --> 00:51:59.130 align:middle line:84% Remember, this is a set of all x's that 00:51:59.130 --> 00:52:02.140 align:middle line:84% get mapped by f squared to alpha to infinity. 00:52:02.140 --> 00:52:06.660 align:middle line:84% And if alpha is negative, then no matter what x is in E, 00:52:06.660 --> 00:52:10.110 align:middle line:84% f squared of x is going to be in alpha to infinity, again, 00:52:10.110 --> 00:52:11.490 align:middle line:90% for alpha less than 0. 00:52:11.490 --> 00:52:15.730 align:middle line:84% So this inverse image equals E, which 00:52:15.730 --> 00:52:17.110 align:middle line:90% is measurable by assumption. 00:52:17.110 --> 00:52:20.460 align:middle line:90% 00:52:20.460 --> 00:52:23.670 align:middle line:84% And then the other less trivial case 00:52:23.670 --> 00:52:30.330 align:middle line:84% is if alpha is bigger than or equal to 0, then f 00:52:30.330 --> 00:52:34.980 align:middle line:84% squared of x is bigger than alpha if and only 00:52:34.980 --> 00:52:39.390 align:middle line:90% if either what? 00:52:39.390 --> 00:52:46.390 align:middle line:84% f of x is bigger than the square root of alpha or f 00:52:46.390 --> 00:52:49.660 align:middle line:84% of x is less than minus square root of alpha. 00:52:49.660 --> 00:52:59.232 align:middle line:84% And therefore-- so, again, this expression here is-- 00:52:59.232 --> 00:53:00.870 align:middle line:90% this expression is expressing. 00:53:00.870 --> 00:53:03.600 align:middle line:84% But this relation here is expressing 00:53:03.600 --> 00:53:07.570 align:middle line:84% x is in the inverse image of alpha to infinity. 00:53:07.570 --> 00:53:16.720 align:middle line:84% So this says that the inverse image of alpha to infinity 00:53:16.720 --> 00:53:19.720 align:middle line:84% is equal to the set of all x's satisfying 00:53:19.720 --> 00:53:22.600 align:middle line:84% this condition on the right of this, if and only if, 00:53:22.600 --> 00:53:25.050 align:middle line:90% which is-- 00:53:25.050 --> 00:53:28.540 align:middle line:84% can be written as x is in the inverse image of-- 00:53:28.540 --> 00:53:37.570 align:middle line:90% 00:53:37.570 --> 00:53:41.110 align:middle line:84% f is bigger than square root of alpha, union, 00:53:41.110 --> 00:53:44.950 align:middle line:84% again, coming from the or, f inverse of-- 00:53:44.950 --> 00:53:50.940 align:middle line:90% 00:53:50.940 --> 00:53:53.070 align:middle line:84% and again, if we have a measurable function, 00:53:53.070 --> 00:53:56.370 align:middle line:84% then not only are these types of the pre-images 00:53:56.370 --> 00:53:58.590 align:middle line:84% of these types of sets measurable, 00:53:58.590 --> 00:54:00.540 align:middle line:84% but the pre-images of these types of sets 00:54:00.540 --> 00:54:01.613 align:middle line:90% are also measurable. 00:54:01.613 --> 00:54:03.030 align:middle line:84% That was the first thing we prove. 00:54:03.030 --> 00:54:06.990 align:middle line:84% And so since each of these pre-images are measurable, 00:54:06.990 --> 00:54:08.560 align:middle line:90% their union is measurable. 00:54:08.560 --> 00:54:10.890 align:middle line:90% So that's measurable. 00:54:10.890 --> 00:54:14.540 align:middle line:90% 00:54:14.540 --> 00:54:17.880 align:middle line:84% So we proved that f squared is measurable. 00:54:17.880 --> 00:54:19.880 align:middle line:84% And now we conclude that f times g 00:54:19.880 --> 00:54:31.140 align:middle line:84% is measurable by a simple identity that f times g-- 00:54:31.140 --> 00:54:37.860 align:middle line:84% this is equal to 1/4 f plus g squared 00:54:37.860 --> 00:54:41.595 align:middle line:90% minus f minus g squared. 00:54:41.595 --> 00:54:45.330 align:middle line:90% 00:54:45.330 --> 00:54:48.980 align:middle line:84% So f times g is equal to this function squared. 00:54:48.980 --> 00:54:51.980 align:middle line:84% If f and g are measurable, their sum is measurable. 00:54:51.980 --> 00:54:53.720 align:middle line:84% And therefore, by what we just proved, 00:54:53.720 --> 00:54:55.520 align:middle line:90% the square is measurable. 00:54:55.520 --> 00:54:58.430 align:middle line:84% And again, over here, this is going to be measurable. 00:54:58.430 --> 00:55:02.510 align:middle line:84% And by scalar multiple of minus 1, that thing's measurable. 00:55:02.510 --> 00:55:06.540 align:middle line:84% Scalar multiple of 1/4 on the outside is fine as well. 00:55:06.540 --> 00:55:13.320 align:middle line:84% So we conclude that this thing is measurable 00:55:13.320 --> 00:55:16.830 align:middle line:84% because every operation in this expression 00:55:16.830 --> 00:55:18.870 align:middle line:84% preserves the function being measurable, 00:55:18.870 --> 00:55:23.610 align:middle line:84% as we proved before this expression. 00:55:23.610 --> 00:55:25.480 align:middle line:84% And that's-- that concludes the proof. 00:55:25.480 --> 00:55:32.980 align:middle line:84% So the sum of two measurable functions is measurable. 00:55:32.980 --> 00:55:34.660 align:middle line:90% Scalar products are measurable. 00:55:34.660 --> 00:55:43.490 align:middle line:84% Products are measurable-- big deal because as far as Riemann 00:55:43.490 --> 00:55:52.997 align:middle line:90% integration goes, that's-- 00:55:52.997 --> 00:55:55.739 align:middle line:90% 00:55:55.739 --> 00:55:57.880 align:middle line:90% that still holds. 00:55:57.880 --> 00:56:01.500 align:middle line:84% So what's something that sets apart 00:56:01.500 --> 00:56:09.130 align:middle line:84% a function being measurable and, eventually, Lebesgue integrable 00:56:09.130 --> 00:56:14.040 align:middle line:90% is that it has better-- 00:56:14.040 --> 00:56:18.160 align:middle line:84% it's closed under taking limits as opposed to being 00:56:18.160 --> 00:56:21.050 align:middle line:90% Riemann integrable. 00:56:21.050 --> 00:56:25.000 align:middle line:84% So this is the following theorem. 00:56:25.000 --> 00:56:31.110 align:middle line:90% 00:56:31.110 --> 00:56:38.130 align:middle line:84% If I have a measurable set and then I 00:56:38.130 --> 00:56:46.250 align:middle line:84% have a sequence of functions, fn, going from minus infinity 00:56:46.250 --> 00:56:48.380 align:middle line:90% to infinity-- 00:56:48.380 --> 00:56:53.510 align:middle line:84% oh, and so I should have said a minute ago-- 00:56:53.510 --> 00:56:56.450 align:middle line:90% I'm just catching myself. 00:56:56.450 --> 00:56:56.950 align:middle line:90% Oh, no. 00:56:56.950 --> 00:56:57.700 align:middle line:90% Everything's fine. 00:56:57.700 --> 00:57:02.980 align:middle line:84% So I just wanted to make sure I had gone to R because f plus g 00:57:02.980 --> 00:57:07.840 align:middle line:84% is only defined going from E to R. 00:57:07.840 --> 00:57:10.480 align:middle line:84% But again, if I have one function that 00:57:10.480 --> 00:57:13.360 align:middle line:84% is finite everywhere and another function that's 00:57:13.360 --> 00:57:15.170 align:middle line:84% an extended real-valued function, 00:57:15.170 --> 00:57:17.320 align:middle line:84% you can check that f plus g is then 00:57:17.320 --> 00:57:21.790 align:middle line:90% going to be measurable as well. 00:57:21.790 --> 00:57:24.880 align:middle line:84% I can't make sense of the sum of two 00:57:24.880 --> 00:57:27.310 align:middle line:84% extended real-valued functions because I 00:57:27.310 --> 00:57:31.510 align:middle line:84% might be in a situation where I have plus infinity minus 00:57:31.510 --> 00:57:36.180 align:middle line:84% infinity, which is an undefined expression. 00:57:36.180 --> 00:57:44.430 align:middle line:84% But what I'm saying is I can take a-- 00:57:44.430 --> 00:57:48.630 align:middle line:84% for all of this, for one of them to be extended real-valued 00:57:48.630 --> 00:57:49.350 align:middle line:90% is fine. 00:57:49.350 --> 00:57:51.840 align:middle line:84% And also, the product of extended real-valued 00:57:51.840 --> 00:57:54.120 align:middle line:84% is fine, although I left it out of the list of rules, 00:57:54.120 --> 00:57:56.760 align:middle line:84% plus infinity times minus infinity 00:57:56.760 --> 00:58:02.325 align:middle line:84% is defined to be minus infinity, and with the usual sign rules. 00:58:02.325 --> 00:58:05.720 align:middle line:90% 00:58:05.720 --> 00:58:09.170 align:middle line:84% So back to the theorem-- if I have a sequence of measurable 00:58:09.170 --> 00:58:12.460 align:middle line:90% functions-- 00:58:12.460 --> 00:58:25.240 align:middle line:84% so then some other functions are measurable. 00:58:25.240 --> 00:58:33.630 align:middle line:84% Then the function g1 of x equal to sup n fn of x-- 00:58:33.630 --> 00:58:36.510 align:middle line:84% so that's now a function defined on E. This is also measurable. 00:58:36.510 --> 00:58:39.193 align:middle line:90% 00:58:39.193 --> 00:58:40.860 align:middle line:84% So I'll just list the functions and then 00:58:40.860 --> 00:58:42.180 align:middle line:90% say they're all measurable. 00:58:42.180 --> 00:58:46.100 align:middle line:90% 00:58:46.100 --> 00:58:52.860 align:middle line:84% g2 of x equals the inf over n fn of x. 00:58:52.860 --> 00:58:56.290 align:middle line:90% 00:58:56.290 --> 00:59:02.420 align:middle line:84% 3 x equals limsup as n goes to infinity 00:59:02.420 --> 00:59:07.640 align:middle line:84% of fn of x, which we'll recall the definition you can write 00:59:07.640 --> 00:59:19.780 align:middle line:84% as the inf over all n of the sup over all k bigger 00:59:19.780 --> 00:59:22.660 align:middle line:90% than or equal to n fk of x. 00:59:22.660 --> 00:59:26.200 align:middle line:90% 00:59:26.200 --> 00:59:33.340 align:middle line:84% And 4-- so I don't know why I'm double-labeling them. 00:59:33.340 --> 00:59:34.450 align:middle line:90% I have 1, 2, 3, and 4. 00:59:34.450 --> 00:59:37.900 align:middle line:84% But I also have 1, 2, 3, and 4 here. 00:59:37.900 --> 00:59:39.850 align:middle line:90% And the liminf of-- 00:59:39.850 --> 00:59:45.560 align:middle line:84% the point-wise liminf of these functions, 00:59:45.560 --> 00:59:51.387 align:middle line:84% which I will recall is equal to the sup over m of inf-- 00:59:51.387 --> 00:59:57.240 align:middle line:90% 00:59:57.240 --> 00:59:59.100 align:middle line:84% these are all measurable functions. 00:59:59.100 --> 01:00:24.390 align:middle line:90% 01:00:24.390 --> 01:00:29.450 align:middle line:90% So let's prove this. 01:00:29.450 --> 01:00:35.950 align:middle line:84% So the proof is not too difficult. 01:00:35.950 --> 01:00:37.710 align:middle line:84% So let's start with the first one. 01:00:37.710 --> 01:00:39.180 align:middle line:84% Actually, the third and the fourth 01:00:39.180 --> 01:00:42.480 align:middle line:84% follow from the first and the second. 01:00:42.480 --> 01:00:55.480 align:middle line:84% But so x is in the inverse image by g1 alpha 01:00:55.480 --> 01:01:10.150 align:middle line:84% to infinity if and only if, of course, sup n fn of x 01:01:10.150 --> 01:01:11.320 align:middle line:90% is greater than alpha. 01:01:11.320 --> 01:01:14.110 align:middle line:84% And this is equivalent to the sup 01:01:14.110 --> 01:01:17.290 align:middle line:84% over all n of fn of x is bigger than alpha 01:01:17.290 --> 01:01:21.490 align:middle line:84% if and only if there exists some n so that fn of x is bigger 01:01:21.490 --> 01:01:22.930 align:middle line:90% than alpha. 01:01:22.930 --> 01:01:26.920 align:middle line:84% If all the fns stay below alpha when evaluated at x, then 01:01:26.920 --> 01:01:29.620 align:middle line:84% the sup is less than or equal to alpha. 01:01:29.620 --> 01:01:33.920 align:middle line:84% So since the sup is the least upper bound-- 01:01:33.920 --> 01:01:40.165 align:middle line:84% so if and only if there exists an n such that-- 01:01:40.165 --> 01:01:49.180 align:middle line:90% 01:01:49.180 --> 01:01:54.625 align:middle line:84% if and only if there exists an n so that x is in fn inverse of-- 01:01:54.625 --> 01:02:00.010 align:middle line:90% 01:02:00.010 --> 01:02:06.970 align:middle line:84% and therefore, we've proven that the inverse image 01:02:06.970 --> 01:02:10.960 align:middle line:84% of the set by g1 is equal to the union 01:02:10.960 --> 01:02:16.330 align:middle line:84% over all n of f inverse f sub n inverse the inverse image 01:02:16.330 --> 01:02:23.300 align:middle line:90% of alpha to infinity. 01:02:23.300 --> 01:02:25.900 align:middle line:84% And now we're assuming the fn's are all measurable. 01:02:25.900 --> 01:02:28.210 align:middle line:90% So each of these is measurable. 01:02:28.210 --> 01:02:30.820 align:middle line:84% It's a countable union of measurable sets. 01:02:30.820 --> 01:02:33.355 align:middle line:84% So it's, again, a measurable set. 01:02:33.355 --> 01:02:39.390 align:middle line:90% 01:02:39.390 --> 01:02:42.570 align:middle line:84% So we've proven that for all alpha, 01:02:42.570 --> 01:02:43.890 align:middle line:90% this is a measurable set. 01:02:43.890 --> 01:02:45.390 align:middle line:90% So g1 is measurable. 01:02:45.390 --> 01:02:49.820 align:middle line:90% 01:02:49.820 --> 01:02:54.860 align:middle line:84% Now, if we go on to g2, it's the same thing. 01:02:54.860 --> 01:02:59.220 align:middle line:84% I can prove that the inverse image of-- 01:02:59.220 --> 01:02:59.720 align:middle line:90% let's see. 01:02:59.720 --> 01:03:00.470 align:middle line:90% What do I do here? 01:03:00.470 --> 01:03:06.420 align:middle line:90% 01:03:06.420 --> 01:03:09.750 align:middle line:84% Now I'm going to include alpha here. 01:03:09.750 --> 01:03:13.450 align:middle line:84% The inverse image of alpha to infinity-- 01:03:13.450 --> 01:03:15.810 align:middle line:84% so this is equal to the requiring 01:03:15.810 --> 01:03:22.230 align:middle line:84% that the inf over fn of x is bigger than or equal to alpha. 01:03:22.230 --> 01:03:24.840 align:middle line:84% This is equivalent to the intersection 01:03:24.840 --> 01:03:36.850 align:middle line:90% now of f inverse of these guys. 01:03:36.850 --> 01:03:42.630 align:middle line:84% Now, each of these is, again, measurable by assumption. 01:03:42.630 --> 01:03:46.260 align:middle line:90% And therefore, it's countable. 01:03:46.260 --> 01:03:48.090 align:middle line:90% Intersection is measurable. 01:03:48.090 --> 01:04:02.450 align:middle line:90% 01:04:02.450 --> 01:04:06.860 align:middle line:84% So we've proven that taking sups and infs of sequences 01:04:06.860 --> 01:04:08.450 align:middle line:90% of functions are measurable. 01:04:08.450 --> 01:04:15.970 align:middle line:90% 01:04:15.970 --> 01:04:22.010 align:middle line:84% But by how the limsup and liminf are defined-- 01:04:22.010 --> 01:04:24.070 align:middle line:84% so now we've proven that for any sequence 01:04:24.070 --> 01:04:27.130 align:middle line:84% of measurable functions, the sup and the infs 01:04:27.130 --> 01:04:29.380 align:middle line:90% are measurable functions. 01:04:29.380 --> 01:04:31.750 align:middle line:90% But the limsup is the sup-- 01:04:31.750 --> 01:04:34.840 align:middle line:84% is this sup first, followed by an inf. 01:04:34.840 --> 01:04:37.060 align:middle line:84% Now, if all of the f's are measurable, 01:04:37.060 --> 01:04:42.660 align:middle line:84% this sup is measurable for all n. 01:04:42.660 --> 01:04:45.336 align:middle line:84% And therefore, this inf is measurable. 01:04:45.336 --> 01:04:50.520 align:middle line:84% So g3 being measurable follows immediately 01:04:50.520 --> 01:04:55.380 align:middle line:84% from proving that sups and infs of measurable functions 01:04:55.380 --> 01:04:58.260 align:middle line:84% are measurable, and the same thing with the liminf. 01:04:58.260 --> 01:05:00.690 align:middle line:84% If f sub k is measurable for all k, 01:05:00.690 --> 01:05:04.080 align:middle line:84% then we've already proven that the infs over the k's-- 01:05:04.080 --> 01:05:07.230 align:middle line:84% bigger than or equal to n doesn't matter-- 01:05:07.230 --> 01:05:10.830 align:middle line:84% is measurable, a measurable function. 01:05:10.830 --> 01:05:12.630 align:middle line:84% And therefore, the sup over the n's 01:05:12.630 --> 01:05:15.120 align:middle line:84% of all these measurable functions 01:05:15.120 --> 01:05:16.470 align:middle line:90% are, again, measurable. 01:05:16.470 --> 01:05:31.350 align:middle line:84% So the fact that g3 and g4 are measurable 01:05:31.350 --> 01:05:41.060 align:middle line:84% follow from the expressions for-- 01:05:41.060 --> 01:05:48.040 align:middle line:90% 01:05:48.040 --> 01:05:57.140 align:middle line:84% and the previous two cases, meaning 01:05:57.140 --> 01:06:00.410 align:middle line:84% that we proved infs and sups of measurable functions 01:06:00.410 --> 01:06:01.760 align:middle line:90% are measurable functions. 01:06:01.760 --> 01:06:06.530 align:middle line:90% 01:06:06.530 --> 01:06:08.360 align:middle line:90% Now, what do we get from this? 01:06:08.360 --> 01:06:12.970 align:middle line:90% 01:06:12.970 --> 01:06:15.990 align:middle line:84% So an immediate theorem is the following. 01:06:15.990 --> 01:06:22.620 align:middle line:90% 01:06:22.620 --> 01:06:26.910 align:middle line:84% If E is measurable, fn goes from-- 01:06:26.910 --> 01:06:36.180 align:middle line:90% 01:06:36.180 --> 01:06:41.520 align:middle line:90% and fn is measurable for all n. 01:06:41.520 --> 01:06:44.610 align:middle line:90% 01:06:44.610 --> 01:06:51.875 align:middle line:84% And they converge pointwise to some function f of x. 01:06:51.875 --> 01:06:58.710 align:middle line:90% 01:06:58.710 --> 01:07:05.720 align:middle line:90% Then f is measurable. 01:07:05.720 --> 01:07:07.983 align:middle line:90% 01:07:07.983 --> 01:07:09.400 align:middle line:84% I shouldn't write it as a theorem. 01:07:09.400 --> 01:07:10.630 align:middle line:84% I should write it as a corollary. 01:07:10.630 --> 01:07:11.130 align:middle line:90% Why? 01:07:11.130 --> 01:07:16.490 align:middle line:84% Because if these converge, then f is equal to the limsup. 01:07:16.490 --> 01:07:17.780 align:middle line:90% It's also equal to the liminf. 01:07:17.780 --> 01:07:21.220 align:middle line:90% But f is equal to the limsup. 01:07:21.220 --> 01:07:31.010 align:middle line:90% So f-- so the proof is one line. 01:07:31.010 --> 01:07:44.290 align:middle line:84% For all x in E, f of x is equal to limsup of fn of x. 01:07:44.290 --> 01:07:52.850 align:middle line:84% And it's also equal to the liminf 01:07:52.850 --> 01:07:55.643 align:middle line:90% if the fns are converging to f. 01:07:55.643 --> 01:08:01.330 align:middle line:90% 01:08:01.330 --> 01:08:04.000 align:middle line:90% You covered this in 18.100. 01:08:04.000 --> 01:08:07.480 align:middle line:84% You have a limit of a sequence if and only 01:08:07.480 --> 01:08:09.940 align:middle line:90% if the limsup equals the liminf. 01:08:09.940 --> 01:08:13.600 align:middle line:84% And this also holds if I include plus or minus infinity 01:08:13.600 --> 01:08:17.649 align:middle line:84% as being possible elements of the extended real numbers 01:08:17.649 --> 01:08:20.154 align:middle line:84% that are in the sequence or the possible limit. 01:08:20.154 --> 01:08:24.920 align:middle line:90% 01:08:24.920 --> 01:08:27.290 align:middle line:84% Since this is measurable by the previous theorem, 01:08:27.290 --> 01:08:28.550 align:middle line:90% f is measurable. 01:08:28.550 --> 01:08:32.220 align:middle line:90% 01:08:32.220 --> 01:08:39.250 align:middle line:84% So this says something that is an indicator that we're 01:08:39.250 --> 01:08:42.670 align:middle line:84% doing something that's or we're building 01:08:42.670 --> 01:08:46.479 align:middle line:84% something that's stronger than being Riemann integrable. 01:08:46.479 --> 01:08:50.770 align:middle line:84% As I've said, the functions which we will define a Lebesgue 01:08:50.770 --> 01:08:57.529 align:middle line:84% integral for will be a certain subset of measurable functions. 01:08:57.529 --> 01:09:00.090 align:middle line:90% 01:09:00.090 --> 01:09:11.120 align:middle line:84% And so taking them as our possible candidates, 01:09:11.120 --> 01:09:16.160 align:middle line:84% let me just make the following remark, which now separates 01:09:16.160 --> 01:09:18.620 align:middle line:84% measurable functions, which are-- again, 01:09:18.620 --> 01:09:20.029 align:middle line:84% are candidates for being Lebesgue 01:09:20.029 --> 01:09:23.029 align:middle line:84% integrable from Riemann integrable 01:09:23.029 --> 01:09:26.930 align:middle line:84% is that this fails if I replace measurable 01:09:26.930 --> 01:09:28.939 align:middle line:90% by Riemann integrable. 01:09:28.939 --> 01:09:35.399 align:middle line:90% 01:09:35.399 --> 01:09:40.510 align:middle line:84% If fn-- let's go from a, b to even R-- 01:09:40.510 --> 01:09:44.600 align:middle line:90% is Riemann integrable. 01:09:44.600 --> 01:09:50.029 align:middle line:90% 01:09:50.029 --> 01:09:56.890 align:middle line:84% And fns converge to f pointwise, which just means this. 01:09:56.890 --> 01:10:00.520 align:middle line:84% For all x in a, b, limit as n goes to infinity 01:10:00.520 --> 01:10:05.540 align:middle line:90% of fn of x equals f of x. 01:10:05.540 --> 01:10:16.890 align:middle line:84% Then f need not be Riemann integrable. 01:10:16.890 --> 01:10:20.650 align:middle line:84% In other words, when I look at Riemann integrable functions, 01:10:20.650 --> 01:10:24.100 align:middle line:84% they're not closed under taking pointwise limits. 01:10:24.100 --> 01:10:26.950 align:middle line:90% 01:10:26.950 --> 01:10:33.340 align:middle line:84% So yet for the Lebesgue integral, 01:10:33.340 --> 01:10:37.450 align:middle line:84% at least for our candidates of Lebesgue measurable functions, 01:10:37.450 --> 01:10:41.350 align:middle line:84% those are closed under taking pointwise limits, 01:10:41.350 --> 01:10:42.460 align:middle line:90% as we just proved. 01:10:42.460 --> 01:10:47.080 align:middle line:84% Now-- which says that we're on track to proving something 01:10:47.080 --> 01:10:49.768 align:middle line:90% that's a-- 01:10:49.768 --> 01:10:52.620 align:middle line:84% that has better qualities than the Riemann integral. 01:10:52.620 --> 01:10:56.610 align:middle line:84% Now, as it turns out, the pointwise limit 01:10:56.610 --> 01:10:59.950 align:middle line:84% of Lebesgue integrable functions need not 01:10:59.950 --> 01:11:01.010 align:middle line:90% be Lebesgue integrable. 01:11:01.010 --> 01:11:04.720 align:middle line:84% But if you add another very minor condition, 01:11:04.720 --> 01:11:08.500 align:middle line:84% the answer is, yes, it is Lebesgue integrable. 01:11:08.500 --> 01:11:13.570 align:middle line:84% But at the very least, we would like for our candidates-- 01:11:13.570 --> 01:11:15.820 align:middle line:84% again, Lebesgue measurable functions-- 01:11:15.820 --> 01:11:19.270 align:middle line:84% to be closed under taking pointwise limits 01:11:19.270 --> 01:11:22.600 align:middle line:84% or, at least, that should indicate to us that we're 01:11:22.600 --> 01:11:26.260 align:middle line:84% doing something that will have better properties than Riemann 01:11:26.260 --> 01:11:27.520 align:middle line:90% integrable functions. 01:11:27.520 --> 01:11:28.840 align:middle line:90% So why is this? 01:11:28.840 --> 01:11:33.220 align:middle line:84% What's a sequence of functions that are Riemann integrable, 01:11:33.220 --> 01:11:34.630 align:middle line:90% but not-- 01:11:34.630 --> 01:11:38.480 align:middle line:84% but their pointwise limit is not Riemann integrable? 01:11:38.480 --> 01:11:47.530 align:middle line:84% So for example, you could take the function fn to be-- 01:11:47.530 --> 01:11:50.730 align:middle line:90% so you know what? 01:11:50.730 --> 01:11:54.540 align:middle line:84% I'm actually going to add this to the assignment. 01:11:54.540 --> 01:11:55.830 align:middle line:90% And it won't be-- 01:11:55.830 --> 01:11:56.580 align:middle line:90% no. 01:11:56.580 --> 01:12:00.360 align:middle line:84% So I'll go ahead and say why this is the case. 01:12:00.360 --> 01:12:03.770 align:middle line:90% 01:12:03.770 --> 01:12:11.360 align:middle line:84% Let's write-- so the rational numbers Q and-- 01:12:11.360 --> 01:12:16.200 align:middle line:90% let's say intersect 0, 1-- 01:12:16.200 --> 01:12:18.497 align:middle line:90% this is a countable set. 01:12:18.497 --> 01:12:20.080 align:middle line:84% It's a subset of the rational numbers. 01:12:20.080 --> 01:12:21.122 align:middle line:90% So this is countable set. 01:12:21.122 --> 01:12:22.290 align:middle line:90% So we can list them. 01:12:22.290 --> 01:12:26.580 align:middle line:84% This is equal to r1, r2, and so on. 01:12:26.580 --> 01:12:29.830 align:middle line:90% 01:12:29.830 --> 01:12:43.380 align:middle line:84% If I define fn of x to be 1 if x is in r1 up to rn and 0 01:12:43.380 --> 01:12:50.540 align:middle line:84% otherwise, now this is a function 01:12:50.540 --> 01:12:53.510 align:middle line:84% which is piecewise continuous on 0, 1. 01:12:53.510 --> 01:12:55.400 align:middle line:90% So it's Riemann integrable. 01:12:55.400 --> 01:13:03.560 align:middle line:90% 01:13:03.560 --> 01:13:07.630 align:middle line:84% But what happens as I take n goes to infinity? 01:13:07.630 --> 01:13:11.640 align:middle line:90% 01:13:11.640 --> 01:13:22.020 align:middle line:84% The pointwise limit of these functions converges 01:13:22.020 --> 01:13:25.920 align:middle line:84% to the indicator function of the rational numbers-- 01:13:25.920 --> 01:13:27.480 align:middle line:90% intersect 0, 1. 01:13:27.480 --> 01:13:29.580 align:middle line:84% And hopefully, this was something 01:13:29.580 --> 01:13:32.940 align:middle line:84% that you checked when you learned 01:13:32.940 --> 01:13:34.260 align:middle line:90% about Riemann integration. 01:13:34.260 --> 01:13:36.720 align:middle line:84% If you just learned about the Riemann 01:13:36.720 --> 01:13:38.460 align:middle line:84% integral of continuous functions, then 01:13:38.460 --> 01:13:39.390 align:middle line:90% that's fine, too. 01:13:39.390 --> 01:13:46.010 align:middle line:84% But anyways, this function is 1 on the rationals and 0 01:13:46.010 --> 01:13:46.880 align:middle line:90% elsewhere. 01:13:46.880 --> 01:13:52.340 align:middle line:84% And-- which is a function which is discontinuous everywhere, 01:13:52.340 --> 01:13:54.910 align:middle line:90% or-- 01:13:54.910 --> 01:13:59.910 align:middle line:84% let's see-- not everywhere, or-- yeah, everywhere. 01:13:59.910 --> 01:14:02.100 align:middle line:84% And you can convince yourself that that's 01:14:02.100 --> 01:14:04.740 align:middle line:84% too crazy of a function to be Riemann integrable 01:14:04.740 --> 01:14:08.820 align:middle line:84% if all you knew was that continuous functions are 01:14:08.820 --> 01:14:09.640 align:middle line:90% Riemann integrable. 01:14:09.640 --> 01:14:12.240 align:middle line:84% But if you learned more about the Riemann integral, 01:14:12.240 --> 01:14:14.190 align:middle line:84% then one of the things that you learned 01:14:14.190 --> 01:14:16.350 align:middle line:84% was that this function is not Riemann integrable. 01:14:16.350 --> 01:14:23.140 align:middle line:90% 01:14:23.140 --> 01:14:28.840 align:middle line:84% So taking the pointwise limit of Riemann integrable functions 01:14:28.840 --> 01:14:31.360 align:middle line:90% may not be Riemann integrable. 01:14:31.360 --> 01:14:35.770 align:middle line:84% However, what we've proven is that the candidates for being 01:14:35.770 --> 01:14:38.860 align:middle line:84% Lebesgue integrable-- namely, those measurable functions-- 01:14:38.860 --> 01:14:41.560 align:middle line:84% are closed under taking pointwise limits, which 01:14:41.560 --> 01:14:50.510 align:middle line:84% is, like I said, an indication that this theory 01:14:50.510 --> 01:14:53.630 align:middle line:84% of the integral that we're building up 01:14:53.630 --> 01:14:56.150 align:middle line:84% is going to be a stronger theory in the sense 01:14:56.150 --> 01:15:02.210 align:middle line:84% that we can prove more things than that of Riemann. 01:15:02.210 --> 01:15:04.910 align:middle line:84% And one of those things we'll prove 01:15:04.910 --> 01:15:11.250 align:middle line:90% is that the space of Riemann-- 01:15:11.250 --> 01:15:15.840 align:middle line:84% I mean, Lebesgue integrable functions with a norm 01:15:15.840 --> 01:15:17.760 align:middle line:84% being the integral of the absolute value 01:15:17.760 --> 01:15:19.655 align:middle line:84% is, in fact, a Banach space as opposed 01:15:19.655 --> 01:15:20.655 align:middle line:90% to the Riemann integral. 01:15:20.655 --> 01:15:32.770 align:middle line:90% 01:15:32.770 --> 01:15:37.590 align:middle line:84% So now let me just make a couple of final statements. 01:15:37.590 --> 01:15:40.740 align:middle line:84% And then we'll call it a day for this lecture. 01:15:40.740 --> 01:15:43.430 align:middle line:90% 01:15:43.430 --> 01:15:52.910 align:middle line:84% So this is just really some terminology 01:15:52.910 --> 01:16:10.410 align:middle line:84% that when I make a statement P of x and I say that with x to-- 01:16:10.410 --> 01:16:14.100 align:middle line:84% an element of a measurable set E-- 01:16:14.100 --> 01:16:30.340 align:middle line:84% If I say this statement holds almost everywhere on E, 01:16:30.340 --> 01:16:32.860 align:middle line:90% and I'll usually say-- 01:16:32.860 --> 01:16:38.820 align:middle line:84% I'll shorten that to just "ae" on E or just 01:16:38.820 --> 01:16:47.710 align:middle line:84% "ae" if I'm deleting even more from what I write-- 01:16:47.710 --> 01:16:55.610 align:middle line:84% this holds almost everywhere if the set where it doesn't-- 01:16:55.610 --> 01:17:09.590 align:middle line:84% where it's-- doesn't hold has measure 0. 01:17:09.590 --> 01:17:13.340 align:middle line:84% So I'm saying-- you may think I'm saying two things here, 01:17:13.340 --> 01:17:15.560 align:middle line:84% that this set is measurable and its measure is 0. 01:17:15.560 --> 01:17:20.880 align:middle line:84% But remember, when we developed the theory of measure, 01:17:20.880 --> 01:17:23.250 align:middle line:84% first thing we learned about measurable sets 01:17:23.250 --> 01:17:26.580 align:middle line:84% is that if it has outer measure equal to 0, then 01:17:26.580 --> 01:17:27.570 align:middle line:90% it's measurable. 01:17:27.570 --> 01:17:30.957 align:middle line:84% And remember, the outer-- the measure of a measurable set 01:17:30.957 --> 01:17:32.040 align:middle line:90% is just its outer measure. 01:17:32.040 --> 01:17:36.820 align:middle line:90% So perhaps I'll say with-- 01:17:36.820 --> 01:17:38.020 align:middle line:90% in parentheses-- 01:17:38.020 --> 01:17:43.650 align:middle line:90% 01:17:43.650 --> 01:17:46.200 align:middle line:84% So a statement holds almost everywhere 01:17:46.200 --> 01:17:53.510 align:middle line:84% if the set where it doesn't hold has a Lebesgue measure 0, which 01:17:53.510 --> 01:17:55.940 align:middle line:84% is equivalent to saying it has outer measure equal to 0 01:17:55.940 --> 01:17:59.240 align:middle line:84% because we proved that sets that have outer measure equal to 0 01:17:59.240 --> 01:18:00.440 align:middle line:90% are Lebesgue measurable. 01:18:00.440 --> 01:18:04.680 align:middle line:90% 01:18:04.680 --> 01:18:09.120 align:middle line:84% And so the final theorem that we'll prove today is that if I 01:18:09.120 --> 01:18:11.640 align:middle line:84% have two functions-- one measurable, 01:18:11.640 --> 01:18:15.370 align:middle line:84% the other differing by that measurable function off a-- 01:18:15.370 --> 01:18:19.830 align:middle line:84% on a set of measure 0-- then the second function 01:18:19.830 --> 01:18:22.200 align:middle line:90% is also measurable. 01:18:22.200 --> 01:18:27.060 align:middle line:84% So somehow, sets of measure 0 don't affect 01:18:27.060 --> 01:18:29.320 align:middle line:90% being a measurable function. 01:18:29.320 --> 01:18:34.530 align:middle line:84% So if fg go from a measurable set E 01:18:34.530 --> 01:18:43.800 align:middle line:84% to the extended real numbers, f is measurable. 01:18:43.800 --> 01:18:50.960 align:middle line:84% And f equals g almost everywhere on E, 01:18:50.960 --> 01:18:53.360 align:middle line:84% meaning the set of x's where f does not 01:18:53.360 --> 01:19:04.610 align:middle line:84% equal g is a set of measures 0 in E. Then g is measurable. 01:19:04.610 --> 01:19:09.110 align:middle line:90% 01:19:09.110 --> 01:19:11.020 align:middle line:84% So if you take a measurable function f, 01:19:11.020 --> 01:19:13.840 align:middle line:84% change it on a set of Lebesgue measure 0, 01:19:13.840 --> 01:19:16.060 align:middle line:84% you still get a measurable function. 01:19:16.060 --> 01:19:19.320 align:middle line:90% 01:19:19.320 --> 01:19:21.070 align:middle line:84% And again, this just follows from the fact 01:19:21.070 --> 01:19:23.950 align:middle line:84% that all sets of Lebesgue measure 01:19:23.950 --> 01:19:28.900 align:middle line:84% 0 or outer measure equal to 0 are Lebesgue measurable. 01:19:28.900 --> 01:19:31.270 align:middle line:84% And then, again, the fact that we know measurable sets 01:19:31.270 --> 01:19:36.460 align:middle line:84% are closed under taking unions and complements-- 01:19:36.460 --> 01:19:43.000 align:middle line:84% so let n be the set of x's in E such that f of x 01:19:43.000 --> 01:19:44.305 align:middle line:90% does not equal g of x. 01:19:44.305 --> 01:19:47.550 align:middle line:90% 01:19:47.550 --> 01:19:51.960 align:middle line:84% Then this is a set of outer measures 0. 01:19:51.960 --> 01:19:53.880 align:middle line:84% And therefore, it's Lebesgue measurable. 01:19:53.880 --> 01:20:06.300 align:middle line:90% 01:20:06.300 --> 01:20:15.750 align:middle line:84% So now if I take an alpha in R and define another set-- 01:20:15.750 --> 01:20:23.610 align:middle line:84% call it N sub alpha-- to be the set of all 01:20:23.610 --> 01:20:32.230 align:middle line:84% x's in N such that g of x is greater than alpha, 01:20:32.230 --> 01:20:39.940 align:middle line:84% this is a subset of N just by definition. 01:20:39.940 --> 01:20:42.460 align:middle line:90% 01:20:42.460 --> 01:20:44.670 align:middle line:84% So this is how I'm defining this set. 01:20:44.670 --> 01:20:54.080 align:middle line:90% 01:20:54.080 --> 01:20:56.390 align:middle line:84% So it has outer measure equal to 0. 01:20:56.390 --> 01:21:02.815 align:middle line:84% And therefore, it's measurable because N has outer measure 0. 01:21:02.815 --> 01:21:12.420 align:middle line:90% 01:21:12.420 --> 01:21:20.870 align:middle line:84% Then if I take the inverse image of alpha to infinity, 01:21:20.870 --> 01:21:27.820 align:middle line:84% this is equal to the inverse image 01:21:27.820 --> 01:21:32.720 align:middle line:84% by f of alpha to infinity intersect, 01:21:32.720 --> 01:21:34.400 align:middle line:90% where they equal each other. 01:21:34.400 --> 01:21:40.930 align:middle line:84% So we'll intersect the complement of N union this set, 01:21:40.930 --> 01:21:41.665 align:middle line:90% N sub alpha. 01:21:41.665 --> 01:21:45.540 align:middle line:90% 01:21:45.540 --> 01:21:50.480 align:middle line:90% Now N has outer measure 0. 01:21:50.480 --> 01:21:52.160 align:middle line:90% So it's Lebesgue measurable. 01:21:52.160 --> 01:21:54.590 align:middle line:84% And therefore, its complement is measurable. 01:21:54.590 --> 01:21:57.380 align:middle line:84% This is measurable by assumption. 01:21:57.380 --> 01:22:00.280 align:middle line:84% And therefore, this intersection is measurable. 01:22:00.280 --> 01:22:02.330 align:middle line:90% And this set here is measurable. 01:22:02.330 --> 01:22:05.710 align:middle line:84% So the union of this measurable set with this measurable set 01:22:05.710 --> 01:22:07.120 align:middle line:90% is, again, measurable. 01:22:07.120 --> 01:22:11.420 align:middle line:84% And we conclude that this is measurable. 01:22:11.420 --> 01:22:22.300 align:middle line:84% So we have defined measurable extended real-valued functions 01:22:22.300 --> 01:22:25.510 align:middle line:84% and proven some properties of them, 01:22:25.510 --> 01:22:28.660 align:middle line:84% the most striking of which being that being measurable 01:22:28.660 --> 01:22:32.610 align:middle line:84% is closed under taking pointwise limits 01:22:32.610 --> 01:22:37.920 align:middle line:84% and that you can change a function on a set of measure 0 01:22:37.920 --> 01:22:40.440 align:middle line:90% and still be measurable. 01:22:40.440 --> 01:22:44.070 align:middle line:84% Next time, we'll extend this notion in a trivial way 01:22:44.070 --> 01:22:48.180 align:middle line:84% to functions which take on complex values, 01:22:48.180 --> 01:22:54.160 align:middle line:84% not extended, just finite complex numbers, 01:22:54.160 --> 01:22:56.950 align:middle line:90% not including the-- 01:22:56.950 --> 01:23:00.990 align:middle line:90% pointed at infinity. 01:23:00.990 --> 01:23:12.710 align:middle line:84% And then we will move on to defining-- 01:23:12.710 --> 01:23:14.105 align:middle line:90% so once we have these-- 01:23:14.105 --> 01:23:18.220 align:middle line:90% so here's the game plan. 01:23:18.220 --> 01:23:20.530 align:middle line:84% So we'll extend this notion of being 01:23:20.530 --> 01:23:24.030 align:middle line:84% measurable to complex-valued functions. 01:23:24.030 --> 01:23:32.770 align:middle line:84% And then we'll show that there's a particular class of functions 01:23:32.770 --> 01:23:36.100 align:middle line:84% called simple functions which, again, are simplest 01:23:36.100 --> 01:23:39.910 align:middle line:84% in the sense that they just take on finitely many values, 01:23:39.910 --> 01:23:44.410 align:middle line:84% and show that those are the universal measurable functions. 01:23:44.410 --> 01:23:48.100 align:middle line:84% And from there, we will then define the Lebesgue integral 01:23:48.100 --> 01:23:52.710 align:middle line:84% of a non-negative measurable function 01:23:52.710 --> 01:23:56.380 align:middle line:84% and prove some properties of that. 01:23:56.380 --> 01:23:59.260 align:middle line:90% And then once we've done that-- 01:23:59.260 --> 01:24:00.950 align:middle line:90% so you can define the-- 01:24:00.950 --> 01:24:04.080 align:middle line:84% an integral of a non-negative-- an arbitrary non-negative 01:24:04.080 --> 01:24:05.220 align:middle line:90% measurable function. 01:24:05.220 --> 01:24:07.020 align:middle line:84% We will then restrict to those functions 01:24:07.020 --> 01:24:13.950 align:middle line:84% which we call Lebesgue integrable, which now can-- 01:24:13.950 --> 01:24:20.820 align:middle line:84% not necessarily non-negative, but will have finite Lebesgue 01:24:20.820 --> 01:24:24.748 align:middle line:84% integral, and then prove those are-- 01:24:24.748 --> 01:24:26.790 align:middle line:84% and then prove some properties of that, including 01:24:26.790 --> 01:24:31.140 align:middle line:84% the big convergence theorems, and then close out 01:24:31.140 --> 01:24:34.980 align:middle line:84% our section on or chapter on measure and integration 01:24:34.980 --> 01:24:39.600 align:middle line:84% by proving that the Lebesgue integrable functions form 01:24:39.600 --> 01:24:42.930 align:middle line:84% a Banach space with a natural norm put 01:24:42.930 --> 01:24:48.040 align:middle line:84% on it as opposed to Riemann integrable functions. 01:24:48.040 --> 01:24:49.910 align:middle line:90% We'll stop there. 01:24:49.910 --> 01:24:57.000 align:middle line:90%