WEBVTT 00:00:00.000 --> 00:00:01.992 align:middle line:90% [SQUEAKING] 00:00:01.992 --> 00:00:03.984 align:middle line:90% [RUSTLING] 00:00:03.984 --> 00:00:21.507 align:middle line:90% [CLICKING] 00:00:21.507 --> 00:00:23.090 align:middle line:84% CASEY RODRIGUEZ: So let's get started. 00:00:23.090 --> 00:00:25.610 align:middle line:84% Last time, at the end of the last lecture, 00:00:25.610 --> 00:00:28.150 align:middle line:90% we introduced outer measure. 00:00:28.150 --> 00:00:30.670 align:middle line:84% So we first discussed what we wanted 00:00:30.670 --> 00:00:33.107 align:middle line:84% to measure, the properties of kind 00:00:33.107 --> 00:00:37.240 align:middle line:84% of a notion measure of subsets of to satisfy. 00:00:37.240 --> 00:00:40.780 align:middle line:84% We wanted it to, first off, be defined for all subsets. 00:00:40.780 --> 00:00:43.120 align:middle line:84% We then wanted the measure of an interval 00:00:43.120 --> 00:00:45.170 align:middle line:84% to be the length of the interval. 00:00:45.170 --> 00:00:51.520 align:middle line:84% We also wanted the measure of a union of disjoint subsets 00:00:51.520 --> 00:00:57.490 align:middle line:84% to be the sum of the measures, and said 00:00:57.490 --> 00:00:59.440 align:middle line:84% we wanted it to be translation invariant, 00:00:59.440 --> 00:01:02.890 align:middle line:84% meaning if I take a set and I shift it by a fixed amount, 00:01:02.890 --> 00:01:05.290 align:middle line:84% then the measure of the shifted set 00:01:05.290 --> 00:01:09.320 align:middle line:84% is the same as the measure of the original set. 00:01:09.320 --> 00:01:12.380 align:middle line:84% But as I said last time, that's impossible. 00:01:12.380 --> 00:01:13.150 align:middle line:90% You can't. 00:01:13.150 --> 00:01:15.040 align:middle line:84% There does not exist such a thing 00:01:15.040 --> 00:01:17.270 align:middle line:84% defined on every subset of real numbers. 00:01:17.270 --> 00:01:25.000 align:middle line:84% So what we're going to do is define-- 00:01:25.000 --> 00:01:29.040 align:middle line:84% or what we're doing right now is defining outer measure, which 00:01:29.040 --> 00:01:33.790 align:middle line:84% will satisfy almost all of the properties 00:01:33.790 --> 00:01:39.580 align:middle line:84% we wanted, and is defined for every subset of R. 00:01:39.580 --> 00:01:42.640 align:middle line:84% And then we'll restrict this outer measure 00:01:42.640 --> 00:01:45.550 align:middle line:84% to a certain class of subsets of R 00:01:45.550 --> 00:01:48.730 align:middle line:84% that are well behaved with respect to measure. 00:01:48.730 --> 00:01:54.580 align:middle line:84% And then we'll get a function, now defined 00:01:54.580 --> 00:01:58.360 align:middle line:84% on a collection of subsets of R, what 00:01:58.360 --> 00:02:00.160 align:middle line:90% we'll call measurable sets. 00:02:00.160 --> 00:02:03.730 align:middle line:84% And this set function, which we call measure, Lebesgue measure, 00:02:03.730 --> 00:02:07.060 align:middle line:84% will satisfy the three properties, the main three 00:02:07.060 --> 00:02:09.750 align:middle line:84% properties we wanted-- that the measure of an interval 00:02:09.750 --> 00:02:11.500 align:middle line:84% is the length of the interval, the measure 00:02:11.500 --> 00:02:15.610 align:middle line:84% of a countable disjoint union of sets 00:02:15.610 --> 00:02:20.680 align:middle line:84% is the sum of the measures, and it's translation invariant. 00:02:20.680 --> 00:02:26.100 align:middle line:84% Now at the end of last time-- so we defined outer measure 00:02:26.100 --> 00:02:27.900 align:middle line:84% and we also proved this theorem that if we 00:02:27.900 --> 00:02:30.510 align:middle line:84% have a countable collection of subsets of R, 00:02:30.510 --> 00:02:33.210 align:middle line:84% then the measure of the union is less than 00:02:33.210 --> 00:02:34.770 align:middle line:84% or equal to the sum of the measures. 00:02:34.770 --> 00:02:37.660 align:middle line:84% Now, what we would like in the end, like I said, 00:02:37.660 --> 00:02:40.650 align:middle line:84% is to be able to have equality whenever 00:02:40.650 --> 00:02:42.600 align:middle line:90% the subsets are disjoint. 00:02:42.600 --> 00:02:45.120 align:middle line:84% Now, this doesn't hold for outer measure, but as we'll see, 00:02:45.120 --> 00:02:49.560 align:middle line:84% once we restrict outer measure to a certain class of subsets, 00:02:49.560 --> 00:02:52.390 align:middle line:84% we will get the property that we want. 00:02:52.390 --> 00:02:58.970 align:middle line:84% Now outer measure almost satisfies 00:02:58.970 --> 00:03:02.420 align:middle line:84% one of those properties we want to have of a measure. 00:03:02.420 --> 00:03:03.890 align:middle line:84% What we're going to verify now is 00:03:03.890 --> 00:03:07.880 align:middle line:84% that outer measure does satisfy one of the properties we wanted 00:03:07.880 --> 00:03:10.490 align:middle line:84% of a measure, namely that the measure of a interval 00:03:10.490 --> 00:03:11.610 align:middle line:90% ought to be its length. 00:03:11.610 --> 00:03:15.820 align:middle line:90% So that's the following-- 00:03:15.820 --> 00:03:25.840 align:middle line:84% if I is an interval, then the measure, the outer measure 00:03:25.840 --> 00:03:30.460 align:middle line:84% of I, is equal to the length of I. So here, the length of I-- 00:03:30.460 --> 00:03:33.760 align:middle line:84% remember, if it's an interval of the form a, b, 00:03:33.760 --> 00:03:37.600 align:middle line:84% with the endpoints included or not, the length of the interval 00:03:37.600 --> 00:03:38.470 align:middle line:90% is b minus a. 00:03:38.470 --> 00:03:44.094 align:middle line:84% If it's an infinite interval, then the length is infinite. 00:03:44.094 --> 00:03:50.630 align:middle line:90% 00:03:50.630 --> 00:03:59.750 align:middle line:84% So the most involved part, or at least the crux of proving this, 00:03:59.750 --> 00:04:03.780 align:middle line:84% is doing the simplest kind of interval, 00:04:03.780 --> 00:04:07.130 align:middle line:84% which is a closed and bounded interval, a, b. 00:04:07.130 --> 00:04:10.430 align:middle line:90% So let's do that case first. 00:04:10.430 --> 00:04:15.125 align:middle line:90% Suppose I is equal to a, b. 00:04:15.125 --> 00:04:24.330 align:middle line:90% 00:04:24.330 --> 00:04:27.960 align:middle line:84% And also, something that I didn't write down here is we 00:04:27.960 --> 00:04:32.880 align:middle line:84% have, immediately from the definition of outer measure-- 00:04:32.880 --> 00:04:34.860 align:middle line:84% so let me just pause that real quick-- 00:04:34.860 --> 00:04:40.350 align:middle line:84% that if A is a subset of B, then the outer measure of A 00:04:40.350 --> 00:04:44.440 align:middle line:84% is less than or equal to the outer measure of B, 00:04:44.440 --> 00:04:49.270 align:middle line:84% simply because if I take any open collection 00:04:49.270 --> 00:04:51.130 align:middle line:84% of open intervals covering B, then 00:04:51.130 --> 00:04:54.010 align:middle line:84% that will be a collection of open intervals covering 00:04:54.010 --> 00:04:58.120 align:middle line:84% A. And since infimums of bigger sets-- 00:04:58.120 --> 00:05:02.875 align:middle line:90% 00:05:02.875 --> 00:05:04.650 align:middle line:84% let's see, which direction is that-- 00:05:04.650 --> 00:05:07.450 align:middle line:90% 00:05:07.450 --> 00:05:11.350 align:middle line:84% since infimums of bigger sets have to decrease, 00:05:11.350 --> 00:05:13.390 align:middle line:84% that gives me the inequality that I want. 00:05:13.390 --> 00:05:18.860 align:middle line:90% 00:05:18.860 --> 00:05:22.460 align:middle line:84% So going back to the proof of this theorem, 00:05:22.460 --> 00:05:23.780 align:middle line:90% let's suppose I equals a, b. 00:05:23.780 --> 00:05:27.350 align:middle line:84% What we're going to show is that the outer measure 00:05:27.350 --> 00:05:29.300 align:middle line:84% is less than or equal to the length of I, 00:05:29.300 --> 00:05:32.300 align:middle line:84% and then the converse-- that the length of I 00:05:32.300 --> 00:05:34.610 align:middle line:84% is less than or equal to the outer measure of I. 00:05:34.610 --> 00:05:37.760 align:middle line:84% So the simpler one is showing the outer measure of I 00:05:37.760 --> 00:05:39.800 align:middle line:84% is less than or equal to the length of I. 00:05:39.800 --> 00:05:45.170 align:middle line:84% So then I is contained in the open interval, 00:05:45.170 --> 00:05:52.420 align:middle line:84% the single open interval, I minus epsilon, 00:05:52.420 --> 00:05:55.900 align:middle line:84% b plus epsilon for all epsilon positive. 00:05:55.900 --> 00:05:58.750 align:middle line:84% And therefore, since the outer measure of I 00:05:58.750 --> 00:06:03.490 align:middle line:84% is the infimum of the sum of lengths of intervals covering 00:06:03.490 --> 00:06:07.570 align:middle line:84% I, this implies that the outer measure of I 00:06:07.570 --> 00:06:19.270 align:middle line:84% is less than or equal to the length of this interval, which 00:06:19.270 --> 00:06:22.210 align:middle line:84% is equal to b minus a plus 2 epsilon 00:06:22.210 --> 00:06:25.120 align:middle line:90% for all epsilon positive. 00:06:25.120 --> 00:06:29.050 align:middle line:84% And therefore, if I have this is less than or equal to b 00:06:29.050 --> 00:06:31.870 align:middle line:84% minus a plus to epsilon for all epsilon positive, 00:06:31.870 --> 00:06:36.680 align:middle line:84% I can then send epsilon to 0 to conclude 00:06:36.680 --> 00:06:39.890 align:middle line:84% that the outer measure of the interval is less than 00:06:39.890 --> 00:06:44.420 align:middle line:84% or equal to b minus a, the length of the interval. 00:06:44.420 --> 00:06:45.590 align:middle line:90% So that's the simple part. 00:06:45.590 --> 00:06:48.420 align:middle line:90% 00:06:48.420 --> 00:07:01.640 align:middle line:84% So now, we're going to show that b minus a is less than 00:07:01.640 --> 00:07:07.785 align:middle line:84% or equal to the outer measure of the interval. 00:07:07.785 --> 00:07:10.890 align:middle line:90% 00:07:10.890 --> 00:07:13.320 align:middle line:84% So also, what this shows is that the outer measure 00:07:13.320 --> 00:07:15.700 align:middle line:90% of this interval is finite. 00:07:15.700 --> 00:07:18.700 align:middle line:90% It's a finite number. 00:07:18.700 --> 00:07:21.750 align:middle line:84% So to show this, what we have to show 00:07:21.750 --> 00:07:26.820 align:middle line:84% is that if I take any collection of open intervals covering I, 00:07:26.820 --> 00:07:29.220 align:middle line:84% then the sum of the lengths of those intervals 00:07:29.220 --> 00:07:31.155 align:middle line:90% is bounded below by b minus a. 00:07:31.155 --> 00:07:38.910 align:middle line:90% 00:07:38.910 --> 00:07:41.250 align:middle line:84% And if I don't write n equals 1 to infinity, 00:07:41.250 --> 00:07:43.050 align:middle line:84% it should be understood that n is always 00:07:43.050 --> 00:07:45.100 align:middle line:90% going from 1 to infinity. 00:07:45.100 --> 00:07:57.380 align:middle line:84% Let I n be a collection of open intervals 00:07:57.380 --> 00:08:07.890 align:middle line:84% such that this interval I, a, b, is contained in the union. 00:08:07.890 --> 00:08:10.710 align:middle line:84% We now want to show that b minus a is less than or equal 00:08:10.710 --> 00:08:12.660 align:middle line:84% to the sum of the measures of the I n's. 00:08:12.660 --> 00:08:15.930 align:middle line:84% Now, the closed and bounded interval, a, b-- this 00:08:15.930 --> 00:08:18.240 align:middle line:90% is a compact set. 00:08:18.240 --> 00:08:22.620 align:middle line:84% So you look back to 100B, which is 00:08:22.620 --> 00:08:25.200 align:middle line:84% covered by a collection of open intervals-- 00:08:25.200 --> 00:08:26.370 align:middle line:90% in other words, open sets. 00:08:26.370 --> 00:08:29.640 align:middle line:90% 00:08:29.640 --> 00:08:33.890 align:middle line:90% The definition of compactness-- 00:08:33.890 --> 00:08:43.130 align:middle line:84% so since this closed and bounded interval is compact, 00:08:43.130 --> 00:08:53.760 align:middle line:84% this is due to a special theorem by Heine and Borel-- 00:08:53.760 --> 00:08:58.170 align:middle line:84% we can find a finite covering of this interval 00:08:58.170 --> 00:09:01.630 align:middle line:84% by finitely many of these intervals. 00:09:01.630 --> 00:09:06.960 align:middle line:84% So I can choose finitely many of these intervals to cover a, b. 00:09:06.960 --> 00:09:08.920 align:middle line:84% There exists a finite collection, 00:09:08.920 --> 00:09:17.310 align:middle line:84% which I will now denote, let's say, J k, k equals 1 to n, 00:09:17.310 --> 00:09:25.640 align:middle line:84% so contained inside this collection of open intervals-- 00:09:25.640 --> 00:09:28.430 align:middle line:84% so these are just finitely many open intervals from this 00:09:28.430 --> 00:09:29.210 align:middle line:90% collection-- 00:09:29.210 --> 00:09:32.290 align:middle line:90% 00:09:32.290 --> 00:09:42.088 align:middle line:84% such that a, b is contained in k equals 1 to n J k. 00:09:42.088 --> 00:09:52.490 align:middle line:90% 00:09:52.490 --> 00:09:56.930 align:middle line:84% So now, I have these open intervals. 00:09:56.930 --> 00:09:59.510 align:middle line:84% And now what I'm going to argue is that I can do this-- 00:09:59.510 --> 00:10:03.680 align:middle line:90% is that here's the plan. 00:10:03.680 --> 00:10:05.840 align:middle line:90% Here's a. 00:10:05.840 --> 00:10:07.460 align:middle line:90% There's b. 00:10:07.460 --> 00:10:08.930 align:middle line:84% What I'm going to argue now to you 00:10:08.930 --> 00:10:17.700 align:middle line:84% is that rearranging how I'm indexing these open intervals, 00:10:17.700 --> 00:10:24.480 align:middle line:84% that I can cover a, b in the following way-- 00:10:24.480 --> 00:10:34.180 align:middle line:90% so this is the first. 00:10:34.180 --> 00:10:37.950 align:middle line:84% So I'm going to argue that we can cover this interval 00:10:37.950 --> 00:10:42.060 align:middle line:84% like this, so that I can choose the first interval to cover 00:10:42.060 --> 00:10:45.880 align:middle line:84% some of a, b, and then maybe it covers all, in which case 00:10:45.880 --> 00:10:47.790 align:middle line:90% I would stop my construction. 00:10:47.790 --> 00:10:54.180 align:middle line:84% If it doesn't cover all of a, b, then I can cover it with J2. 00:10:54.180 --> 00:10:58.530 align:middle line:90% So now, I have a2, b2. 00:10:58.530 --> 00:11:01.140 align:middle line:84% And if that doesn't cover all of a, b, 00:11:01.140 --> 00:11:06.270 align:middle line:84% I will still have to choose some more intervals to cover it. 00:11:06.270 --> 00:11:10.380 align:middle line:90% And eventually, I'll get to-- 00:11:10.380 --> 00:11:14.430 align:middle line:84% at least in this picture, since I can't draw n intervals-- 00:11:14.430 --> 00:11:17.640 align:middle line:84% I will be able to cover a, b in such a way 00:11:17.640 --> 00:11:32.610 align:middle line:84% that these are kind of linked together, 00:11:32.610 --> 00:11:37.100 align:middle line:84% so that these intervals are linked. 00:11:37.100 --> 00:11:39.980 align:middle line:90% And then I cover all of a, b. 00:11:39.980 --> 00:11:41.240 align:middle line:90% Now, why is that great? 00:11:41.240 --> 00:11:44.870 align:middle line:84% Well, then, because the sum of the lengths of these intervals 00:11:44.870 --> 00:11:47.510 align:middle line:84% is going to be bigger than or equal to-- 00:11:47.510 --> 00:11:50.330 align:middle line:84% the sum of the lengths of these intervals is going to be what? 00:11:50.330 --> 00:11:56.000 align:middle line:84% b3 minus a1, which is bigger than or equal to b minus a. 00:11:56.000 --> 00:12:00.530 align:middle line:84% And that would give me the lower bound that I desire. 00:12:00.530 --> 00:12:02.370 align:middle line:84% So now, let me argue that we can do this-- 00:12:02.370 --> 00:12:05.960 align:middle line:90% that this picture is correct. 00:12:05.960 --> 00:12:10.090 align:middle line:90% 00:12:10.090 --> 00:12:12.250 align:middle line:90% Since a is in this-- 00:12:12.250 --> 00:12:15.870 align:middle line:90% 00:12:15.870 --> 00:12:18.930 align:middle line:84% it's in a, b, so it has to be in this finite union 00:12:18.930 --> 00:12:20.730 align:middle line:90% of open intervals-- 00:12:20.730 --> 00:12:32.810 align:middle line:84% there exists k1 such that a is in J of k1. 00:12:32.810 --> 00:12:36.270 align:middle line:84% So one of these intervals, a has to be in it. 00:12:36.270 --> 00:12:54.210 align:middle line:84% And by rearranging the intervals, 00:12:54.210 --> 00:13:05.710 align:middle line:84% I can assume that k1 equals 1, i.e., a is in J1, which is a1, 00:13:05.710 --> 00:13:06.210 align:middle line:90% b1. 00:13:06.210 --> 00:13:14.420 align:middle line:90% 00:13:14.420 --> 00:13:18.860 align:middle line:84% Now, it's possible that this whole interval 00:13:18.860 --> 00:13:24.650 align:middle line:84% covers a, b, in which case I would stop the construction. 00:13:24.650 --> 00:13:25.775 align:middle line:90% Otherwise, I continue. 00:13:25.775 --> 00:13:28.580 align:middle line:90% 00:13:28.580 --> 00:13:34.870 align:middle line:84% If b1 is less than or equal to b, 00:13:34.870 --> 00:13:42.190 align:middle line:84% then b1 is certainly in this interval, which implies 00:13:42.190 --> 00:13:45.935 align:middle line:90% there exists a J k sub 2-- 00:13:45.935 --> 00:13:49.410 align:middle line:90% 00:13:49.410 --> 00:13:50.760 align:middle line:90% let me write it this way-- 00:13:50.760 --> 00:14:00.940 align:middle line:84% there exists k sub 2 such that b1 is in j sub k sub 2. 00:14:00.940 --> 00:14:09.740 align:middle line:84% And again, by rearranging, we can 00:14:09.740 --> 00:14:18.650 align:middle line:90% assume that k sub 2 equals 2. 00:14:18.650 --> 00:14:20.690 align:middle line:84% So by re-arranging the remaining intervals, 00:14:20.690 --> 00:14:22.730 align:middle line:90% I can assume k2 is 2. 00:14:22.730 --> 00:14:25.430 align:middle line:84% So if in the case, like I've drawn in the picture, 00:14:25.430 --> 00:14:30.380 align:middle line:84% b1 is not bigger than b, then I can find another interval J2 00:14:30.380 --> 00:14:33.050 align:middle line:90% that contains b1. 00:14:33.050 --> 00:14:45.750 align:middle line:90% 00:14:45.750 --> 00:14:56.780 align:middle line:90% So b1 is in J2, which is a2, b2. 00:14:56.780 --> 00:15:05.250 align:middle line:84% And I will just continue this until the endpoint of one 00:15:05.250 --> 00:15:07.830 align:middle line:90% of these intervals passes b. 00:15:07.830 --> 00:15:11.070 align:middle line:84% The first instance when one of these intervals passes b, I 00:15:11.070 --> 00:15:12.660 align:middle line:90% stop this process. 00:15:12.660 --> 00:15:16.320 align:middle line:84% And it has to stop because all of these intervals do cover b, 00:15:16.320 --> 00:15:17.910 align:middle line:84% so it has to occur at some point. 00:15:17.910 --> 00:15:19.870 align:middle line:84% And there's only finitely many intervals. 00:15:19.870 --> 00:15:23.770 align:middle line:90% 00:15:23.770 --> 00:15:29.080 align:middle line:84% And I'm going to write if b2 is less than or equal to b, 00:15:29.080 --> 00:15:31.730 align:middle line:84% but I'm going to put dot, dot, dot there. 00:15:31.730 --> 00:15:34.180 align:middle line:84% So what have we done by this argument? 00:15:34.180 --> 00:15:38.800 align:middle line:90% 00:15:38.800 --> 00:15:51.370 align:middle line:90% Thus, we conclude that-- 00:15:51.370 --> 00:15:53.470 align:middle line:84% maybe that's a little bit bad notation 00:15:53.470 --> 00:15:55.510 align:middle line:90% because I have n there. 00:15:55.510 --> 00:15:58.090 align:middle line:84% So let's make this a capital N. Sorry about that. 00:15:58.090 --> 00:16:04.760 align:middle line:90% 00:16:04.760 --> 00:16:08.930 align:middle line:84% Make that a capital N, since that little n appears also 00:16:08.930 --> 00:16:10.325 align:middle line:90% to index the I sub n's. 00:16:10.325 --> 00:16:13.490 align:middle line:90% 00:16:13.490 --> 00:16:21.490 align:middle line:84% So thus, we conclude that there exists a k with 1 less than 00:16:21.490 --> 00:16:27.340 align:middle line:84% or equal to k less than or equal to n such 00:16:27.340 --> 00:16:33.160 align:middle line:90% that three things hold. 00:16:33.160 --> 00:16:38.260 align:middle line:84% For all k equals 1 to k minus 1, we 00:16:38.260 --> 00:16:51.820 align:middle line:84% have that b k is less than or equal to b and a k 00:16:51.820 --> 00:16:58.270 align:middle line:84% plus 1 is less than b k is less than b k plus 1. 00:16:58.270 --> 00:17:03.310 align:middle line:84% So think of this as a condition where we have not yet covered 00:17:03.310 --> 00:17:04.854 align:middle line:90% the entire interval a, b. 00:17:04.854 --> 00:17:07.510 align:middle line:90% 00:17:07.510 --> 00:17:12.155 align:middle line:84% For k equals 1, this was basically 00:17:12.155 --> 00:17:13.405 align:middle line:90% the argument I just gave here. 00:17:13.405 --> 00:17:16.510 align:middle line:90% 00:17:16.510 --> 00:17:20.240 align:middle line:84% And two, that this has to stop at some point, 00:17:20.240 --> 00:17:32.260 align:middle line:84% so that b is less than b capital K. So what I drew here-- 00:17:32.260 --> 00:17:39.830 align:middle line:84% so the picture is for the K is K equals 3. 00:17:39.830 --> 00:17:53.000 align:middle line:90% 00:17:53.000 --> 00:17:55.873 align:middle line:84% So we have this K, and I drew the picture over there for b3. 00:17:55.873 --> 00:17:57.290 align:middle line:84% Now I'm going to show that the sum 00:17:57.290 --> 00:18:01.100 align:middle line:84% of the links of the intervals-- and this 00:18:01.100 --> 00:18:04.713 align:middle line:84% is just a finite collection from the bigger 00:18:04.713 --> 00:18:06.380 align:middle line:84% collection of intervals-- is now bounded 00:18:06.380 --> 00:18:08.660 align:middle line:84% by b minus a, which is kind of clear from the picture 00:18:08.660 --> 00:18:09.160 align:middle line:90% that I drew. 00:18:09.160 --> 00:18:11.120 align:middle line:84% But we still have to write stuff down. 00:18:11.120 --> 00:18:12.830 align:middle line:90% This is not Topology. 00:18:12.830 --> 00:18:19.760 align:middle line:84% Then sum of length of I sub n's-- 00:18:19.760 --> 00:18:22.160 align:middle line:84% this is certainly less than or equal to the sum 00:18:22.160 --> 00:18:34.010 align:middle line:84% over this finite subcollection of intervals, lengths of J k. 00:18:34.010 --> 00:18:39.380 align:middle line:84% And this is less than or equal to sum k, K-- 00:18:39.380 --> 00:18:42.290 align:middle line:84% now here, this capital K is coming from over there-- length 00:18:42.290 --> 00:18:44.810 align:middle line:90% of J k. 00:18:44.810 --> 00:18:46.040 align:middle line:90% And now what is this? 00:18:46.040 --> 00:18:54.170 align:middle line:84% This is equal to b K minus a K plus b K 00:18:54.170 --> 00:19:00.050 align:middle line:84% minus 1 minus a K minus 1 plus, and then all the way 00:19:00.050 --> 00:19:11.140 align:middle line:90% down to b1 minus a1. 00:19:11.140 --> 00:19:16.400 align:middle line:90% 00:19:16.400 --> 00:19:23.150 align:middle line:84% But here's the trick-- remember, each of the b K minus 1's, 00:19:23.150 --> 00:19:27.470 align:middle line:84% they lie ahead of a K. So the index here is shifted just 00:19:27.470 --> 00:19:28.140 align:middle line:90% by a little bit. 00:19:28.140 --> 00:19:37.010 align:middle line:84% But this says that the previous b is in front of the next a. 00:19:37.010 --> 00:19:45.110 align:middle line:84% So I can collect terms and write this as b K plus b K minus 1 00:19:45.110 --> 00:19:54.960 align:middle line:84% minus a K plus b K minus 2 minus a K minus 1. 00:19:54.960 --> 00:19:58.120 align:middle line:84% So I just borrowed a b coming after this term, 00:19:58.120 --> 00:20:03.870 align:middle line:84% so on, and so on, until I get b1 minus a2. 00:20:03.870 --> 00:20:08.710 align:middle line:90% And then I'm stuck with a a1. 00:20:08.710 --> 00:20:13.630 align:middle line:84% And by the second condition over here, everything in parentheses 00:20:13.630 --> 00:20:14.840 align:middle line:90% is non-negative. 00:20:14.840 --> 00:20:21.710 align:middle line:84% So this is bigger than or equal to b K minus a1. 00:20:21.710 --> 00:20:22.700 align:middle line:90% And what do we know? 00:20:22.700 --> 00:20:27.130 align:middle line:84% We know b K is bigger than b, and we know a1 is less than a 00:20:27.130 --> 00:20:29.920 align:middle line:90% because a is in a1, b1. 00:20:29.920 --> 00:20:34.360 align:middle line:84% So this length is bigger than b minus a. 00:20:34.360 --> 00:20:40.630 align:middle line:90% 00:20:40.630 --> 00:20:44.740 align:middle line:84% And since we've shown that the sum of lengths of intervals 00:20:44.740 --> 00:20:49.180 align:middle line:84% covering I is bigger than or equal to b minus a no matter 00:20:49.180 --> 00:20:50.770 align:middle line:84% what the collection of intervals is, 00:20:50.770 --> 00:20:52.600 align:middle line:84% then the infimum has to be bigger than 00:20:52.600 --> 00:20:54.130 align:middle line:90% or equal to b minus a. 00:20:54.130 --> 00:21:00.520 align:middle line:84% And we conclude that the outer measure of this interval a, b, 00:21:00.520 --> 00:21:04.165 align:middle line:84% is bigger than or equal to b minus a. 00:21:04.165 --> 00:21:07.900 align:middle line:84% And therefore, I have both sides of the inequality I want, 00:21:07.900 --> 00:21:09.550 align:middle line:90% and therefore, I have equality. 00:21:09.550 --> 00:21:24.200 align:middle line:90% 00:21:24.200 --> 00:21:27.660 align:middle line:84% Now, this was for a closed and bounded interval. 00:21:27.660 --> 00:21:33.030 align:middle line:84% But this essentially gives us the result for any interval. 00:21:33.030 --> 00:21:58.820 align:middle line:84% So if I use any finite interval of the form a, b, a, b, 00:21:58.820 --> 00:22:12.440 align:middle line:84% not including a, open, and then for all 00:22:12.440 --> 00:22:15.360 align:middle line:84% epsilon positive, at least sufficiently small, 00:22:15.360 --> 00:22:16.230 align:middle line:90% what do I have? 00:22:16.230 --> 00:22:23.240 align:middle line:84% I have that a plus epsilon, b minus epsilon 00:22:23.240 --> 00:22:32.300 align:middle line:84% is contained in I, which is contained in a minus epsilon b 00:22:32.300 --> 00:22:34.110 align:middle line:90% plus epsilon. 00:22:34.110 --> 00:22:35.880 align:middle line:90% I is one of these intervals. 00:22:35.880 --> 00:22:37.400 align:middle line:84% So if I just make it a little fatter 00:22:37.400 --> 00:22:42.295 align:middle line:84% and take that closed interval containing it, I get this side. 00:22:42.295 --> 00:22:43.670 align:middle line:84% And then if I just kind of shrink 00:22:43.670 --> 00:22:46.640 align:middle line:84% the interval a little bit and take the closed interval, 00:22:46.640 --> 00:22:51.200 align:middle line:84% this guy will be contained inside of it. 00:22:51.200 --> 00:22:53.990 align:middle line:84% And therefore, the measure of this interval 00:22:53.990 --> 00:22:56.150 align:middle line:84% has to be less than or equal to the outer measure 00:22:56.150 --> 00:22:57.650 align:middle line:84% of this interval, which is less than 00:22:57.650 --> 00:23:00.090 align:middle line:84% or equal to the outer measure of this interval. 00:23:00.090 --> 00:23:01.640 align:middle line:90% And I get that-- 00:23:01.640 --> 00:23:04.590 align:middle line:84% now, the measure of this interval-- 00:23:04.590 --> 00:23:05.465 align:middle line:90% I'll just write this. 00:23:05.465 --> 00:23:21.980 align:middle line:90% 00:23:21.980 --> 00:23:31.150 align:middle line:84% And this over here is b minus a minus 2 epsilon 00:23:31.150 --> 00:23:36.200 align:middle line:84% is less than or equal to the outer measure of the interval, 00:23:36.200 --> 00:23:39.610 align:middle line:84% which is less than or equal to the outer measure of b 00:23:39.610 --> 00:23:41.990 align:middle line:90% minus a plus 2 epsilon. 00:23:41.990 --> 00:23:45.670 align:middle line:90% 00:23:45.670 --> 00:23:48.190 align:middle line:84% And this holds for all epsilon positive. 00:23:48.190 --> 00:23:51.600 align:middle line:90% 00:23:51.600 --> 00:23:53.820 align:middle line:84% Recall that I was starting off for all epsilon 00:23:53.820 --> 00:23:56.490 align:middle line:84% positive, at least sufficiently small depending on b and a, 00:23:56.490 --> 00:23:59.160 align:middle line:84% smaller than the difference of b and a. 00:23:59.160 --> 00:24:02.142 align:middle line:90% 00:24:02.142 --> 00:24:07.350 align:middle line:84% And therefore, if I send epsilon to 0, I get b minus a 00:24:07.350 --> 00:24:11.070 align:middle line:84% is less than or equal to the outer measure of this interval, 00:24:11.070 --> 00:24:15.240 align:middle line:84% this finite interval is less than or equal to-- 00:24:15.240 --> 00:24:16.860 align:middle line:90% what why do I have this here? 00:24:16.860 --> 00:24:19.990 align:middle line:90% 00:24:19.990 --> 00:24:23.730 align:middle line:84% Is less than or equal to b minus a. 00:24:23.730 --> 00:24:31.110 align:middle line:84% And therefore, the outer measure of any one 00:24:31.110 --> 00:24:34.140 align:middle line:84% of these finite intervals is the length of the interval. 00:24:34.140 --> 00:24:41.190 align:middle line:84% And then I'm going to leave it to you as an exercise, 00:24:41.190 --> 00:24:50.670 align:middle line:84% which is not difficult. If I is equal to R negative infinity 00:24:50.670 --> 00:25:02.210 align:middle line:90% to a, a infinity, or include-- 00:25:02.210 --> 00:25:10.550 align:middle line:90% 00:25:10.550 --> 00:25:18.300 align:middle line:84% then the outer measure of this interval is infinity. 00:25:18.300 --> 00:25:24.430 align:middle line:90% 00:25:24.430 --> 00:25:27.280 align:middle line:84% In other words, I cannot ever cover these intervals 00:25:27.280 --> 00:25:31.830 align:middle line:84% by a countable collection of intervals whose sum of lengths 00:25:31.830 --> 00:25:35.200 align:middle line:90% is a finite number. 00:25:35.200 --> 00:25:37.280 align:middle line:90% Again, this is not hard. 00:25:37.280 --> 00:25:50.920 align:middle line:90% 00:25:50.920 --> 00:25:55.090 align:middle line:84% So we get kind of a nice, little fact 00:25:55.090 --> 00:25:59.530 align:middle line:84% from the theorem we proved last time, which 00:25:59.530 --> 00:26:03.130 align:middle line:84% was that if I have a collection of subsets of R, 00:26:03.130 --> 00:26:04.990 align:middle line:84% then the measure of the union is less 00:26:04.990 --> 00:26:06.698 align:middle line:84% than or equal to the sum of the measures, 00:26:06.698 --> 00:26:10.120 align:middle line:84% and this theorem, which says that the measure of an interval 00:26:10.120 --> 00:26:14.140 align:middle line:84% is the length of the interval, which is the following-- 00:26:14.140 --> 00:26:25.090 align:middle line:90% 00:26:25.090 --> 00:26:35.860 align:middle line:84% for every subset of R and epsilon positive, 00:26:35.860 --> 00:26:45.060 align:middle line:84% there exists an open set O such that A is contained in O 00:26:45.060 --> 00:26:49.590 align:middle line:84% and the measure of A, which is less than 00:26:49.590 --> 00:26:51.300 align:middle line:84% or equal to the measure of O simply 00:26:51.300 --> 00:26:54.450 align:middle line:84% because A is contained in O, is less than 00:26:54.450 --> 00:27:01.420 align:middle line:84% or equal to the outer measure of A plus epsilon. 00:27:01.420 --> 00:27:06.760 align:middle line:84% So somehow, the outer measure of subsets 00:27:06.760 --> 00:27:13.450 align:middle line:84% can be approximately measured by the outer measure of open sets. 00:27:13.450 --> 00:27:16.170 align:middle line:84% So another way to say this is, at least with respect 00:27:16.170 --> 00:27:19.950 align:middle line:84% to outer measure, every set can be 00:27:19.950 --> 00:27:23.250 align:middle line:84% approximated by an open set, an outer measure. 00:27:23.250 --> 00:27:25.650 align:middle line:90% And what's the proof of this? 00:27:25.650 --> 00:27:34.930 align:middle line:84% So this is clear if the outer measure is infinite. 00:27:34.930 --> 00:27:37.420 align:middle line:84% Then I just take O to be the entire real number line. 00:27:37.420 --> 00:27:40.020 align:middle line:90% 00:27:40.020 --> 00:27:45.565 align:middle line:90% So suppose it is finite. 00:27:45.565 --> 00:27:49.710 align:middle line:90% 00:27:49.710 --> 00:28:05.720 align:middle line:84% Let I n be a collection of open intervals such 00:28:05.720 --> 00:28:12.053 align:middle line:90% that they cover A. 00:28:12.053 --> 00:28:14.230 align:middle line:84% And remember, the outer measure is the infimum. 00:28:14.230 --> 00:28:16.840 align:middle line:84% So I can get arbitrarily close to that infimum 00:28:16.840 --> 00:28:21.580 align:middle line:84% by summing lengths of certain collections of open intervals. 00:28:21.580 --> 00:28:32.340 align:middle line:84% And sum of the links is less than or equal to, 00:28:32.340 --> 00:28:36.305 align:middle line:84% if you like, the outer measure of A plus epsilon. 00:28:36.305 --> 00:28:40.745 align:middle line:90% 00:28:40.745 --> 00:28:42.370 align:middle line:84% So I should have said at the beginning, 00:28:42.370 --> 00:28:44.500 align:middle line:84% let A be a subset of R and epsilon be positive, 00:28:44.500 --> 00:28:48.712 align:middle line:90% but you understand. 00:28:48.712 --> 00:28:51.950 align:middle line:90% 00:28:51.950 --> 00:28:54.040 align:middle line:84% And so what I do is I just take O to be 00:28:54.040 --> 00:28:57.790 align:middle line:90% this union of open intervals. 00:28:57.790 --> 00:29:06.140 align:middle line:90% 00:29:06.140 --> 00:29:10.290 align:middle line:84% So O is a union of open intervals. 00:29:10.290 --> 00:29:12.600 align:middle line:84% Each of these open intervals is an open set. 00:29:12.600 --> 00:29:16.020 align:middle line:84% And you should know that the union of open sets, 00:29:16.020 --> 00:29:18.210 align:middle line:84% any collection of open sets, is, again, an open set. 00:29:18.210 --> 00:29:19.020 align:middle line:90% So this is open. 00:29:19.020 --> 00:29:24.640 align:middle line:90% 00:29:24.640 --> 00:29:31.270 align:middle line:84% A is contained in O because A is contained in the union. 00:29:31.270 --> 00:29:38.990 align:middle line:84% And the outer measure of O, which 00:29:38.990 --> 00:29:46.580 align:middle line:84% is equal to the outer measure of this union of open intervals 00:29:46.580 --> 00:29:49.850 align:middle line:84% is, by the theorem we proved last time, which 00:29:49.850 --> 00:29:52.820 align:middle line:84% I stated with the definition, or when 00:29:52.820 --> 00:29:55.970 align:middle line:84% I recall the definition is less than or equal to the sum 00:29:55.970 --> 00:30:01.700 align:middle line:84% of the outer measures of the intervals, which we just 00:30:01.700 --> 00:30:06.140 align:middle line:84% proved is the length of these intervals. 00:30:06.140 --> 00:30:09.440 align:middle line:84% And how we chose these intervals, remember, 00:30:09.440 --> 00:30:11.570 align:middle line:84% is so that we have this condition. 00:30:11.570 --> 00:30:13.265 align:middle line:84% And this is less than or equal to-- 00:30:13.265 --> 00:30:17.830 align:middle line:90% 00:30:17.830 --> 00:30:22.420 align:middle line:84% so with respect to outer measure, 00:30:22.420 --> 00:30:30.580 align:middle line:84% every set can be approximated by the approximation with respect 00:30:30.580 --> 00:30:34.060 align:middle line:84% to outer measure by an open set, by a suitable open set. 00:30:34.060 --> 00:30:50.750 align:middle line:90% 00:30:50.750 --> 00:30:53.312 align:middle line:90% So we've defined outer measure. 00:30:53.312 --> 00:30:54.770 align:middle line:84% We've proven some properties of it. 00:30:54.770 --> 00:30:58.610 align:middle line:84% Now, we're in a position to at least define measurable sets. 00:30:58.610 --> 00:31:05.500 align:middle line:90% 00:31:05.500 --> 00:31:08.310 align:middle line:84% And I should say these are Lebesgue measurable sets. 00:31:08.310 --> 00:31:10.910 align:middle line:90% 00:31:10.910 --> 00:31:30.630 align:middle line:84% So a subset of real numbers is Lebesgue measurable-- 00:31:30.630 --> 00:31:36.330 align:middle line:84% so this is a new piece of terminology-- 00:31:36.330 --> 00:31:46.690 align:middle line:84% if for all subsets of R, if I look at the outer measure of A, 00:31:46.690 --> 00:31:50.410 align:middle line:84% this is equal to the outer measure 00:31:50.410 --> 00:31:58.700 align:middle line:84% of A intersect E plus the outer measure of E A intersect E 00:31:58.700 --> 00:32:00.860 align:middle line:90% complement. 00:32:00.860 --> 00:32:05.150 align:middle line:84% This is the definition of being a Lebesgue measurable set. 00:32:05.150 --> 00:32:09.350 align:middle line:84% So in some sense, E is a well-behaved set 00:32:09.350 --> 00:32:16.500 align:middle line:84% if it cuts A into reasonable sets. 00:32:16.500 --> 00:32:18.030 align:middle line:90% What's the best way to say that? 00:32:18.030 --> 00:32:19.440 align:middle line:90% I guess that's OK. 00:32:19.440 --> 00:32:22.370 align:middle line:90% 00:32:22.370 --> 00:32:27.200 align:middle line:84% So a set is measurable if and only if for all A, 00:32:27.200 --> 00:32:29.045 align:middle line:90% we have this equality here. 00:32:29.045 --> 00:32:32.140 align:middle line:90% 00:32:32.140 --> 00:32:35.500 align:middle line:90% So let me make a few remarks. 00:32:35.500 --> 00:32:42.010 align:middle line:90% 00:32:42.010 --> 00:32:48.380 align:middle line:84% First off, this is the left-hand side. 00:32:48.380 --> 00:32:52.490 align:middle line:84% No matter what A and E are, the left-hand side 00:32:52.490 --> 00:32:55.400 align:middle line:84% is always less than or equal to the right-hand side 00:32:55.400 --> 00:33:01.625 align:middle line:84% by the theorem we have up there, that the outer measure 00:33:01.625 --> 00:33:03.500 align:middle line:84% of the union is less than or equal to the sum 00:33:03.500 --> 00:33:04.410 align:middle line:90% of the measures. 00:33:04.410 --> 00:33:20.710 align:middle line:84% So since for all A E, A is contained in, or in fact, 00:33:20.710 --> 00:33:34.310 align:middle line:84% equal to A intersect E union A intersect E complement, 00:33:34.310 --> 00:33:37.760 align:middle line:84% we get that the outer measure of A 00:33:37.760 --> 00:33:41.450 align:middle line:84% is always less than or equal to the outer measure 00:33:41.450 --> 00:33:47.360 align:middle line:84% of A intersect E plus the outer measure of A intersect 00:33:47.360 --> 00:33:52.430 align:middle line:84% the complemented of E. This is regardless of A and E. I mean, 00:33:52.430 --> 00:33:55.820 align:middle line:84% if E is measurable or not, this always holds. 00:33:55.820 --> 00:34:04.120 align:middle line:84% So since this always holds, we could state being measurable 00:34:04.120 --> 00:34:07.330 align:middle line:84% just by, instead of equality, satisfying 00:34:07.330 --> 00:34:09.159 align:middle line:90% one of the inequalities. 00:34:09.159 --> 00:34:18.340 align:middle line:84% Thus, E is measurable if for all subsets of R, 00:34:18.340 --> 00:34:32.100 align:middle line:84% we have that A is less than or equal to the outer measure 00:34:32.100 --> 00:34:36.810 align:middle line:84% of A. Because like I said, we know no matter 00:34:36.810 --> 00:34:39.600 align:middle line:90% what A and E are, this holds. 00:34:39.600 --> 00:34:43.139 align:middle line:84% So if I want equality, then I have to have this as less than 00:34:43.139 --> 00:34:43.900 align:middle line:90% or equal to that. 00:34:43.900 --> 00:34:47.580 align:middle line:84% So E is measurable if and only if this is less than 00:34:47.580 --> 00:34:56.550 align:middle line:84% or equal to the outer measure of A. 00:34:56.550 --> 00:35:01.080 align:middle line:90% So now we have the silliest-- 00:35:01.080 --> 00:35:04.670 align:middle line:90% 00:35:04.670 --> 00:35:07.520 align:middle line:84% well, let's not state it as an example, 00:35:07.520 --> 00:35:10.500 align:middle line:84% but let's we can state it as a theorem. 00:35:10.500 --> 00:35:14.440 align:middle line:90% 00:35:14.440 --> 00:35:17.410 align:middle line:84% It will be a silly theorem, one which I will not 00:35:17.410 --> 00:35:18.970 align:middle line:90% write the actual proof of. 00:35:18.970 --> 00:35:24.070 align:middle line:90% But the empty set-- 00:35:24.070 --> 00:35:26.940 align:middle line:84% so again, I've told you before that 00:35:26.940 --> 00:35:30.400 align:middle line:84% when I'm writing on the board, writing in my notes, 00:35:30.400 --> 00:35:34.230 align:middle line:90% I typically shorten words. 00:35:34.230 --> 00:35:38.900 align:middle line:84% So "measurable" you will see written as "mble" 00:35:38.900 --> 00:35:42.860 align:middle line:84% throughout the notes, and also when I write on the board. 00:35:42.860 --> 00:35:45.290 align:middle line:84% I forget the word for what it means to shorten words. 00:35:45.290 --> 00:35:49.930 align:middle line:90% 00:35:49.930 --> 00:35:57.940 align:middle line:90% So the empty set is measurable. 00:35:57.940 --> 00:36:01.630 align:middle line:90% R is measurable. 00:36:01.630 --> 00:36:10.750 align:middle line:84% And we have the fact that a subset of R 00:36:10.750 --> 00:36:19.400 align:middle line:84% is measurable if and only if its complement is measurable 00:36:19.400 --> 00:36:24.140 align:middle line:84% because either one of these definitions-- 00:36:24.140 --> 00:36:28.880 align:middle line:84% remember, by what we've proven about outer measure, 00:36:28.880 --> 00:36:33.170 align:middle line:84% this equality is equivalent to requiring this inequality. 00:36:33.170 --> 00:36:35.700 align:middle line:90% 00:36:35.700 --> 00:36:38.780 align:middle line:84% These are symmetric in E and E complement. 00:36:38.780 --> 00:36:41.400 align:middle line:90% 00:36:41.400 --> 00:36:44.040 align:middle line:84% So E is measurable if and only if its complement 00:36:44.040 --> 00:36:44.880 align:middle line:90% is measurable. 00:36:44.880 --> 00:36:56.560 align:middle line:90% 00:36:56.560 --> 00:37:01.420 align:middle line:84% All right, so these are kind of the stupidest measurable 00:37:01.420 --> 00:37:05.600 align:middle line:90% sets you could have. 00:37:05.600 --> 00:37:07.730 align:middle line:90% Again, the empty set-- why? 00:37:07.730 --> 00:37:10.070 align:middle line:84% Because this is then empty and the outer measure 00:37:10.070 --> 00:37:12.350 align:middle line:90% of the empty set is 0. 00:37:12.350 --> 00:37:13.880 align:middle line:84% And then over here, I would just get 00:37:13.880 --> 00:37:16.610 align:middle line:84% the measure of A intersect the empty set complement, which 00:37:16.610 --> 00:37:20.810 align:middle line:84% is R. So I get the outer measure of A 00:37:20.810 --> 00:37:22.460 align:middle line:84% is equal to the outer measure of A. 00:37:22.460 --> 00:37:24.500 align:middle line:84% And since the empty set is measurable, 00:37:24.500 --> 00:37:26.450 align:middle line:84% its complement R is also measurable. 00:37:26.450 --> 00:37:29.010 align:middle line:90% 00:37:29.010 --> 00:37:39.620 align:middle line:84% So let's do some non-stupid examples of measurable sets, 00:37:39.620 --> 00:37:43.560 align:middle line:84% still kind of trivial because they don't make up very much. 00:37:43.560 --> 00:37:58.800 align:middle line:84% But if a set has outer measure 0, then it's measurable. 00:37:58.800 --> 00:38:06.090 align:middle line:90% 00:38:06.090 --> 00:38:07.470 align:middle line:90% So how do we prove this? 00:38:07.470 --> 00:38:11.400 align:middle line:84% Again, we just need to prove this inequality here 00:38:11.400 --> 00:38:18.990 align:middle line:84% for every subset of R. So let A be a subset of R. Then 00:38:18.990 --> 00:38:22.440 align:middle line:84% A intersect E, this is contained in E. 00:38:22.440 --> 00:38:26.550 align:middle line:84% And therefore, since we know outer measure is-- 00:38:26.550 --> 00:38:29.952 align:middle line:84% so the fancy word for when A is a subset of B, 00:38:29.952 --> 00:38:31.410 align:middle line:84% the outer measure of A is less than 00:38:31.410 --> 00:38:32.910 align:middle line:84% or equal to the outer measure of B-- 00:38:32.910 --> 00:38:35.610 align:middle line:84% the fancy word for that is "monotonicity." 00:38:35.610 --> 00:38:38.280 align:middle line:84% But since the outer measure is monotonic, 00:38:38.280 --> 00:38:41.130 align:middle line:84% that tells me that the outer measure of E 00:38:41.130 --> 00:38:44.760 align:middle line:84% is less than or equal to the A intersect E is less than 00:38:44.760 --> 00:38:48.090 align:middle line:84% or equal to the outer measure of E. 00:38:48.090 --> 00:38:52.080 align:middle line:84% And since that is 0, that implies 00:38:52.080 --> 00:38:58.023 align:middle line:84% that the outer measure of A intersect E is 0. 00:38:58.023 --> 00:39:00.440 align:middle line:84% Because the outer measure is always a non-negative number. 00:39:00.440 --> 00:39:04.830 align:middle line:90% 00:39:04.830 --> 00:39:12.890 align:middle line:84% Thus, the outer measure of A intersect E 00:39:12.890 --> 00:39:16.880 align:middle line:84% plus the outer measure of A intersect E 00:39:16.880 --> 00:39:19.580 align:middle line:84% complement-- this is 0, so this is 00:39:19.580 --> 00:39:24.200 align:middle line:84% equal to the outer measure of A intersect E complement. 00:39:24.200 --> 00:39:28.130 align:middle line:84% A intersect E complement is contained in A. 00:39:28.130 --> 00:39:31.585 align:middle line:84% So the outer measure of A intersect E complement 00:39:31.585 --> 00:39:32.960 align:middle line:84% is going to be less than or equal 00:39:32.960 --> 00:39:35.720 align:middle line:84% to the outer measure of A, again, 00:39:35.720 --> 00:39:38.645 align:middle line:84% because this set is contained within A. 00:39:38.645 --> 00:39:42.950 align:middle line:90% And thus, E is measurable. 00:39:42.950 --> 00:39:52.540 align:middle line:90% 00:39:52.540 --> 00:39:57.780 align:middle line:84% So right now, after two theorems, 00:39:57.780 --> 00:40:01.320 align:middle line:84% we've shown that the uninteresting sets are 00:40:01.320 --> 00:40:03.030 align:middle line:84% measurable, if by "uninteresting" 00:40:03.030 --> 00:40:08.620 align:middle line:84% we mean having very little measure, 00:40:08.620 --> 00:40:10.540 align:middle line:84% even though I haven't defined measure yet. 00:40:10.540 --> 00:40:11.980 align:middle line:90% I've just defined outer measure. 00:40:11.980 --> 00:40:22.010 align:middle line:84% I'm using these terms interchangeably, unfortunately, 00:40:22.010 --> 00:40:25.850 align:middle line:84% but you'll see why either by the end of this class 00:40:25.850 --> 00:40:26.960 align:middle line:90% or the end of next class. 00:40:26.960 --> 00:40:38.360 align:middle line:90% 00:40:38.360 --> 00:40:44.610 align:middle line:84% But so far, we don't have very many interesting examples 00:40:44.610 --> 00:40:45.750 align:middle line:90% of measurable sets. 00:40:45.750 --> 00:40:48.430 align:middle line:90% 00:40:48.430 --> 00:40:50.260 align:middle line:84% What we will show is that there's 00:40:50.260 --> 00:40:53.480 align:middle line:84% a lot of interesting measurable sets. 00:40:53.480 --> 00:40:58.040 align:middle line:84% In fact, every open set is measurable, 00:40:58.040 --> 00:41:02.720 align:middle line:84% and since if an open set is measurable 00:41:02.720 --> 00:41:06.350 align:middle line:84% its complement is measurable, and the complements 00:41:06.350 --> 00:41:08.750 align:middle line:84% of open sets are closed sets, then we 00:41:08.750 --> 00:41:13.100 align:middle line:84% have that every closed set is measurable, as well. 00:41:13.100 --> 00:41:16.040 align:middle line:84% But in fact, it's much richer than that, as we'll see. 00:41:16.040 --> 00:41:18.730 align:middle line:90% 00:41:18.730 --> 00:41:22.720 align:middle line:84% You can take a countable collection of open sets 00:41:22.720 --> 00:41:24.040 align:middle line:90% and take their intersection. 00:41:24.040 --> 00:41:26.590 align:middle line:84% That's not necessarily an open set, 00:41:26.590 --> 00:41:29.650 align:middle line:84% so it's not clear if it's measurable by what I just said. 00:41:29.650 --> 00:41:36.030 align:middle line:84% But it turns out that this intersection of open sets 00:41:36.030 --> 00:41:37.200 align:middle line:90% will be measurable. 00:41:37.200 --> 00:41:39.180 align:middle line:84% And again, by taking complements, 00:41:39.180 --> 00:41:42.240 align:middle line:84% you get unions of closed sets, which are not necessarily 00:41:42.240 --> 00:41:46.000 align:middle line:90% closed, are also measurable. 00:41:46.000 --> 00:41:51.440 align:middle line:84% So when I learned measure theory, 00:41:51.440 --> 00:41:56.620 align:middle line:84% my instructor told me that if you can write down the set, 00:41:56.620 --> 00:41:59.800 align:middle line:90% chances are it's measurable. 00:41:59.800 --> 00:42:02.440 align:middle line:84% If you can sit down and write down 00:42:02.440 --> 00:42:06.910 align:middle line:84% a union of intersections of complements of so on and so on 00:42:06.910 --> 00:42:09.850 align:middle line:84% of some basic sets, then that's measurable. 00:42:09.850 --> 00:42:13.405 align:middle line:84% And we'll see why that's true shortly. 00:42:13.405 --> 00:42:17.610 align:middle line:90% 00:42:17.610 --> 00:42:22.140 align:middle line:84% Now, before we get to showing everything I just told you, 00:42:22.140 --> 00:42:26.610 align:middle line:84% we need some general facts about measurable sets, 00:42:26.610 --> 00:42:31.950 align:middle line:84% about the structure of the collection of measurable sets. 00:42:31.950 --> 00:42:35.370 align:middle line:90% 00:42:35.370 --> 00:42:39.840 align:middle line:84% Right now, we just know that the collection of measurable sets 00:42:39.840 --> 00:42:46.370 align:middle line:84% includes the empty set R and the sets that 00:42:46.370 --> 00:42:55.350 align:middle line:84% have outer measure equal to 0, which I see I didn't write. 00:42:55.350 --> 00:42:57.860 align:middle line:84% This is also the danger of lecturing 00:42:57.860 --> 00:43:01.070 align:middle line:84% in an empty classroom is that if I make a mistake on the board, 00:43:01.070 --> 00:43:03.520 align:middle line:90% there's no one to correct me. 00:43:03.520 --> 00:43:08.782 align:middle line:84% So next, prove the following theorem about measurable sets-- 00:43:08.782 --> 00:43:10.240 align:middle line:84% that if I have two measurable sets, 00:43:10.240 --> 00:43:12.910 align:middle line:90% their union is measurable. 00:43:12.910 --> 00:43:26.690 align:middle line:84% If E1, E2 are measurable, then their union, E1 union E2, 00:43:26.690 --> 00:43:27.590 align:middle line:90% is measurable. 00:43:27.590 --> 00:43:32.070 align:middle line:90% 00:43:32.070 --> 00:43:39.152 align:middle line:84% So again, we have to verify that inequality 00:43:39.152 --> 00:43:40.360 align:middle line:90% to show that it's measurable. 00:43:40.360 --> 00:43:58.870 align:middle line:84% So let A be a subset of R. So since E2 is measurable, 00:43:58.870 --> 00:44:08.230 align:middle line:84% we get that the measure of A intersect the complement of E1 00:44:08.230 --> 00:44:14.920 align:middle line:84% is equal to the outer measure of A intersect E1 complement 00:44:14.920 --> 00:44:25.000 align:middle line:84% intersect E2 plus the outer measure of E of A intersect E1 00:44:25.000 --> 00:44:28.396 align:middle line:84% complement intersect E2 complement. 00:44:28.396 --> 00:44:31.450 align:middle line:90% 00:44:31.450 --> 00:44:34.040 align:middle line:84% You may be asking, where the hell did this come from? 00:44:34.040 --> 00:44:35.030 align:middle line:90% Why do I care? 00:44:35.030 --> 00:44:38.230 align:middle line:84% Well, it's because I'm getting something. 00:44:38.230 --> 00:44:47.270 align:middle line:84% Remember, this is equal to, by De Morgan's law, 00:44:47.270 --> 00:44:51.170 align:middle line:84% the complement of the union of E1 and E2. 00:44:51.170 --> 00:44:54.710 align:middle line:84% So I want to somehow have some relation involving 00:44:54.710 --> 00:44:57.950 align:middle line:84% the complement of E1 union E2 intersect A. 00:44:57.950 --> 00:45:00.320 align:middle line:84% Because again, I'm trying to show 00:45:00.320 --> 00:45:02.720 align:middle line:84% that the measure of A intersect E1 union 00:45:02.720 --> 00:45:07.790 align:middle line:84% E2 plus the outer measure of A intersect the complement of E1 00:45:07.790 --> 00:45:11.880 align:middle line:84% union E2, which is exactly this term, equals the measure of A. 00:45:11.880 --> 00:45:16.550 align:middle line:84% So it looks like this relation was just grabbed out 00:45:16.550 --> 00:45:17.420 align:middle line:90% of left field. 00:45:17.420 --> 00:45:20.645 align:middle line:84% But that's the thinking behind why you would care. 00:45:20.645 --> 00:45:27.510 align:middle line:90% 00:45:27.510 --> 00:45:41.720 align:middle line:84% So now, A, if I take A and intersect it with E1 union E2, 00:45:41.720 --> 00:45:56.040 align:middle line:84% this is equal to A intersect E1 union A intersect E2. 00:45:56.040 --> 00:46:07.880 align:middle line:90% 00:46:07.880 --> 00:46:11.450 align:middle line:84% Now, everything from here that also 00:46:11.450 --> 00:46:15.420 align:middle line:84% has something in common with E1 is contained in this set. 00:46:15.420 --> 00:46:21.590 align:middle line:84% So this union is actually equal to A intersect E1 union A 00:46:21.590 --> 00:46:27.800 align:middle line:84% intersect E2 intersect E1 complement. 00:46:27.800 --> 00:46:31.010 align:middle line:90% 00:46:31.010 --> 00:46:34.850 align:middle line:84% Now, this appeared here, so you can kind of 00:46:34.850 --> 00:46:37.700 align:middle line:84% see maybe some magic is going to start to happen in a minute 00:46:37.700 --> 00:46:39.680 align:middle line:84% when we start taking the measure. 00:46:39.680 --> 00:46:44.710 align:middle line:90% So what do we get? 00:46:44.710 --> 00:46:52.770 align:middle line:84% We get that the measure of A intersect E1 union 00:46:52.770 --> 00:46:57.540 align:middle line:84% E2, which is this side, this is less than 00:46:57.540 --> 00:47:04.800 align:middle line:84% or equal to the outer measure of A-- 00:47:04.800 --> 00:47:09.510 align:middle line:84% so this is equal to this union, so the outer measure of this 00:47:09.510 --> 00:47:12.120 align:middle line:84% is less than or equal to the sum of the outer measure of this 00:47:12.120 --> 00:47:13.635 align:middle line:90% and the outer measure of this. 00:47:13.635 --> 00:47:26.770 align:middle line:90% 00:47:26.770 --> 00:47:32.570 align:middle line:84% Now, we use the fact that E1 is measurable. 00:47:32.570 --> 00:47:37.580 align:middle line:84% So since E1 is measurable, the outer measure of A intersect E1 00:47:37.580 --> 00:47:41.060 align:middle line:84% is equal to the outer measure of A minus 00:47:41.060 --> 00:47:45.230 align:middle line:84% the outer measure of A intersect E1 complement. 00:47:45.230 --> 00:47:51.960 align:middle line:90% 00:47:51.960 --> 00:47:54.360 align:middle line:84% Or I should say that's backwards. 00:47:54.360 --> 00:48:07.840 align:middle line:90% 00:48:07.840 --> 00:48:10.495 align:middle line:84% And then plus, still, this outer measure here. 00:48:10.495 --> 00:48:27.100 align:middle line:90% 00:48:27.100 --> 00:48:32.150 align:middle line:84% But now, we're in good shape because what do we have? 00:48:32.150 --> 00:48:35.305 align:middle line:84% We have this term here appearing here. 00:48:35.305 --> 00:48:41.120 align:middle line:90% 00:48:41.120 --> 00:48:47.090 align:middle line:84% I also have this term here is appearing here. 00:48:47.090 --> 00:48:51.460 align:middle line:84% So subtracting this over here and subtracting 00:48:51.460 --> 00:48:56.100 align:middle line:84% that over there, I get that the right-hand side 00:48:56.100 --> 00:49:00.870 align:middle line:84% is equal to the outer measure of A minus 00:49:00.870 --> 00:49:04.980 align:middle line:84% the outer measure of A intersect. 00:49:04.980 --> 00:49:11.550 align:middle line:84% And I'll just rewrite this intersection of complements 00:49:11.550 --> 00:49:16.320 align:middle line:84% as the complement of the union by De Morgan's law. 00:49:16.320 --> 00:49:19.950 align:middle line:84% And remember, I started off with the outer measure 00:49:19.950 --> 00:49:23.850 align:middle line:84% of A intersect E1 union E2, and I 00:49:23.850 --> 00:49:26.370 align:middle line:84% showed its less than or equal to the outer measure 00:49:26.370 --> 00:49:30.570 align:middle line:84% of A minus the outer measure of A intersect the complement, 00:49:30.570 --> 00:49:38.100 align:middle line:84% and therefore, the outer measure of A intersect E1, E2 00:49:38.100 --> 00:49:43.470 align:middle line:84% plus the outer measure of A intersect E1 00:49:43.470 --> 00:49:51.130 align:middle line:84% union E2 complement is less than or equal to the outer measure 00:49:51.130 --> 00:49:57.630 align:middle line:84% of A. And that's what we wanted to prove, 00:49:57.630 --> 00:50:02.310 align:middle line:84% and therefore, E1 union E2 is measurable. 00:50:02.310 --> 00:50:25.220 align:middle line:90% 00:50:25.220 --> 00:50:34.290 align:middle line:84% Now, if you can do something for two things, 00:50:34.290 --> 00:50:35.880 align:middle line:84% you can do something for n things, 00:50:35.880 --> 00:50:37.530 align:middle line:84% typically, by an induction argument. 00:50:37.530 --> 00:50:43.050 align:middle line:84% So the previous theorem implies the following-- 00:50:43.050 --> 00:50:53.410 align:middle line:84% that if E1 up to E n are measurable, then 00:50:53.410 --> 00:51:01.397 align:middle line:84% this finite union k equals 1 to n E k is measurable. 00:51:01.397 --> 00:51:02.480 align:middle line:90% And how do you prove this? 00:51:02.480 --> 00:51:03.720 align:middle line:90% You prove it by induction. 00:51:03.720 --> 00:51:09.610 align:middle line:90% So proof by induction-- 00:51:09.610 --> 00:51:12.360 align:middle line:90% 00:51:12.360 --> 00:51:15.700 align:middle line:90% when n equals 1-- 00:51:15.700 --> 00:51:17.125 align:middle line:90% the base case-- this is clear. 00:51:17.125 --> 00:51:22.080 align:middle line:90% 00:51:22.080 --> 00:51:28.640 align:middle line:84% Suppose-- so call this claim star-- 00:51:28.640 --> 00:51:38.790 align:middle line:84% suppose this claim star holds for n equals m. 00:51:38.790 --> 00:51:40.860 align:middle line:84% Now we want to show it holds, that this 00:51:40.860 --> 00:51:44.010 align:middle line:84% implies that the claim of the theorem 00:51:44.010 --> 00:51:45.585 align:middle line:90% holds with n equals m plus 1. 00:51:45.585 --> 00:52:01.180 align:middle line:90% 00:52:01.180 --> 00:52:12.670 align:middle line:84% So let E1 E m plus 1 be measurable. 00:52:12.670 --> 00:52:16.430 align:middle line:90% 00:52:16.430 --> 00:52:22.930 align:middle line:84% Then the union k equals 1 to m plus 1 of E k 00:52:22.930 --> 00:52:27.880 align:middle line:84% is equal to union k equals 1 to m 00:52:27.880 --> 00:52:35.350 align:middle line:90% of E k union or E sub m plus 1. 00:52:35.350 --> 00:52:38.740 align:middle line:84% Now, by the induction hypothesis, 00:52:38.740 --> 00:52:42.580 align:middle line:84% a collection of m measurable sets, 00:52:42.580 --> 00:52:44.200 align:middle line:84% their union is measurable, so this 00:52:44.200 --> 00:52:54.260 align:middle line:84% is measurable by the induction hypothesis. 00:52:54.260 --> 00:52:55.910 align:middle line:84% And we're assuming this is measurable. 00:52:55.910 --> 00:52:57.500 align:middle line:84% And then by the previous theorem, 00:52:57.500 --> 00:53:00.360 align:middle line:84% the union of two measurable sets is measurable. 00:53:00.360 --> 00:53:04.650 align:middle line:90% 00:53:04.650 --> 00:53:06.960 align:middle line:90% And that proves the theorem. 00:53:06.960 --> 00:53:15.000 align:middle line:90% 00:53:15.000 --> 00:53:23.370 align:middle line:84% So up to this point, we've shown two basic things 00:53:23.370 --> 00:53:25.200 align:middle line:90% about measurable sets-- 00:53:25.200 --> 00:53:26.220 align:middle line:90% really three. 00:53:26.220 --> 00:53:29.070 align:middle line:90% First off, it's nonempty. 00:53:29.070 --> 00:53:31.770 align:middle line:84% It's a nonempty collection of measurable sets. 00:53:31.770 --> 00:53:35.760 align:middle line:84% We've shown that a set is measurable if 00:53:35.760 --> 00:53:38.280 align:middle line:84% and only if its complement is measurable. 00:53:38.280 --> 00:53:46.760 align:middle line:84% And we've also shown that finite unions of measurable sets 00:53:46.760 --> 00:53:50.100 align:middle line:90% are again measurable. 00:53:50.100 --> 00:54:02.400 align:middle line:84% And therefore, if I look at the collection 00:54:02.400 --> 00:54:06.360 align:middle line:84% of all measurable sets, this has a very special structure 00:54:06.360 --> 00:54:09.520 align:middle line:84% or a very general type of structure, 00:54:09.520 --> 00:54:12.250 align:middle line:84% which I'm now going to elaborate on. 00:54:12.250 --> 00:54:18.270 align:middle line:84% So let's take a pause here about measurable sets because now, 00:54:18.270 --> 00:54:20.760 align:middle line:84% we're going to say a few general things 00:54:20.760 --> 00:54:29.230 align:middle line:84% about certain classes of sets or certain classes of collections 00:54:29.230 --> 00:54:29.830 align:middle line:90% of sets. 00:54:29.830 --> 00:54:38.415 align:middle line:84% So let me make a definition here. 00:54:38.415 --> 00:54:41.180 align:middle line:90% 00:54:41.180 --> 00:54:53.695 align:middle line:90% A nonempty collection of sets-- 00:54:53.695 --> 00:54:58.210 align:middle line:90% 00:54:58.210 --> 00:55:00.780 align:middle line:84% so this is a collection of subsets of R, 00:55:00.780 --> 00:55:04.380 align:middle line:84% so it's a subset of the power set of R-- 00:55:04.380 --> 00:55:05.355 align:middle line:90% is an algebra. 00:55:05.355 --> 00:55:10.180 align:middle line:90% 00:55:10.180 --> 00:55:12.940 align:middle line:84% So some of you have taken Algebra or are taking Algebra. 00:55:12.940 --> 00:55:17.670 align:middle line:84% So there is a notion of what an algebra is in Algebra. 00:55:17.670 --> 00:55:20.070 align:middle line:84% Here, we say a collection of sets 00:55:20.070 --> 00:55:21.720 align:middle line:84% is an algebra if in some sense it's 00:55:21.720 --> 00:55:29.650 align:middle line:84% closed under taking complements and finite unions. 00:55:29.650 --> 00:55:32.500 align:middle line:84% If two conditions are satisfied-- 00:55:32.500 --> 00:55:36.380 align:middle line:90% 00:55:36.380 --> 00:55:40.010 align:middle line:84% if E is in the algebra, then its complement 00:55:40.010 --> 00:55:54.960 align:middle line:84% is in the algebra, and two, if I take finitely many elements 00:55:54.960 --> 00:55:58.110 align:middle line:84% in the algebra and finitely many subsets 00:55:58.110 --> 00:56:05.540 align:middle line:84% from this algebra of subsets, then this union 00:56:05.540 --> 00:56:07.280 align:middle line:90% is also in this algebra. 00:56:07.280 --> 00:56:11.480 align:middle line:90% 00:56:11.480 --> 00:56:14.090 align:middle line:90% Now, this is for finite unions. 00:56:14.090 --> 00:56:19.730 align:middle line:84% You can ask, what about infinite unions? 00:56:19.730 --> 00:56:24.480 align:middle line:84% That's not equivalent to finite unions, meaning that's 00:56:24.480 --> 00:56:26.310 align:middle line:90% a stronger condition to impose. 00:56:26.310 --> 00:56:30.210 align:middle line:84% So we have a special name for collections 00:56:30.210 --> 00:56:33.600 align:middle line:90% of subsets of that type. 00:56:33.600 --> 00:56:42.560 align:middle line:84% We say an algebra A is a sigma algebra-- 00:56:42.560 --> 00:56:45.980 align:middle line:84% sigma for, if you like, sum, because we really 00:56:45.980 --> 00:56:51.450 align:middle line:90% have a summation in mind-- 00:56:51.450 --> 00:56:54.050 align:middle line:90% 00:56:54.050 --> 00:57:04.390 align:middle line:84% if also the following, stronger condition is satisfied-- 00:57:04.390 --> 00:57:21.310 align:middle line:84% if I have a countable collection of subsets in A, 00:57:21.310 --> 00:57:26.950 align:middle line:84% so a countable collection of elements of the algebra, 00:57:26.950 --> 00:57:37.710 align:middle line:84% then their union is in the algebra. 00:57:37.710 --> 00:57:40.820 align:middle line:84% So an algebra is closed under taking 00:57:40.820 --> 00:57:43.220 align:middle line:90% complements and finite unions. 00:57:43.220 --> 00:57:47.030 align:middle line:84% A sigma algebra is closed under complements, 00:57:47.030 --> 00:57:50.700 align:middle line:84% and also the stronger condition of taking countable unions. 00:57:50.700 --> 00:57:54.200 align:middle line:84% So of course, this condition implies this condition-- 00:57:54.200 --> 00:57:55.880 align:middle line:90% actually, that's not yet clear. 00:57:55.880 --> 00:57:58.410 align:middle line:90% 00:57:58.410 --> 00:58:00.670 align:middle line:90% We'll see. 00:58:00.670 --> 00:58:03.270 align:middle line:84% Although it's not in the definition, 00:58:03.270 --> 00:58:06.420 align:middle line:84% we'll see in just a second that, in fact, this 00:58:06.420 --> 00:58:10.050 align:middle line:84% implies that the empty set has to be a member of the algebra. 00:58:10.050 --> 00:58:13.140 align:middle line:84% And therefore, this condition implies this condition 00:58:13.140 --> 00:58:15.030 align:middle line:84% if you just take finitely many of these 00:58:15.030 --> 00:58:20.595 align:middle line:84% and then take them to be empty after some finite n. 00:58:20.595 --> 00:58:26.360 align:middle line:90% 00:58:26.360 --> 00:58:28.880 align:middle line:84% So that's the definition of an algebra. 00:58:28.880 --> 00:58:31.400 align:middle line:84% That's the definition of a sigma algebra. 00:58:31.400 --> 00:58:41.765 align:middle line:90% 00:58:41.765 --> 00:58:47.870 align:middle line:84% Let me give a few examples real quick or a few remarks 00:58:47.870 --> 00:58:53.250 align:middle line:90% about this. 00:58:53.250 --> 00:59:07.810 align:middle line:84% First off, by De Morgan's laws, which tells you the complement 00:59:07.810 --> 00:59:10.720 align:middle line:84% of a union is an intersect of the intersection 00:59:10.720 --> 00:59:21.580 align:middle line:84% of the complements, we get that E1 E n in the algebra implies 00:59:21.580 --> 00:59:33.130 align:middle line:84% that E1 intersect-- so their intersection, 00:59:33.130 --> 00:59:35.830 align:middle line:84% so this is just for an algebra now-- 00:59:35.830 --> 00:59:45.620 align:middle line:84% their intersection, which is equal to the complement 00:59:45.620 --> 00:59:48.850 align:middle line:84% of the union of their complements. 00:59:48.850 --> 00:59:52.360 align:middle line:84% So since each of these is in the algebra, 00:59:52.360 --> 00:59:54.160 align:middle line:84% the complement is in the algebra. 00:59:54.160 --> 00:59:56.740 align:middle line:84% And therefore, the finite union is in the algebra. 00:59:56.740 --> 00:59:59.180 align:middle line:84% And therefore, the complement of that is in the algebra. 00:59:59.180 --> 01:00:06.410 align:middle line:84% So not only is an algebra closed under taking 01:00:06.410 --> 01:00:08.510 align:middle line:84% complements and finite unions, it's 01:00:08.510 --> 01:00:11.690 align:middle line:84% also closed under taking intersections, 01:00:11.690 --> 01:00:12.980 align:middle line:90% finite intersections. 01:00:12.980 --> 01:00:17.760 align:middle line:90% 01:00:17.760 --> 01:00:30.740 align:middle line:84% So thus, if I take some element from my nonempty algebra, 01:00:30.740 --> 01:00:37.160 align:middle line:84% the empty set, which is equal to E intersect E complement, 01:00:37.160 --> 01:00:38.855 align:middle line:90% that's in the algebra. 01:00:38.855 --> 01:00:43.240 align:middle line:90% 01:00:43.240 --> 01:00:48.100 align:middle line:84% So let me pause for a minute and write this a little carefully. 01:00:48.100 --> 01:00:50.620 align:middle line:84% If I take an element of the algebra which 01:00:50.620 --> 01:00:57.430 align:middle line:84% is supposed to be nonempty, then the empty set, which 01:00:57.430 --> 01:01:00.190 align:middle line:84% is equal to E intersect its complement, 01:01:00.190 --> 01:01:01.660 align:middle line:90% that's in the algebra. 01:01:01.660 --> 01:01:02.770 align:middle line:90% That's in the algebra. 01:01:02.770 --> 01:01:05.320 align:middle line:84% The intersection of two things in the algebra 01:01:05.320 --> 01:01:08.090 align:middle line:90% is in the algebra. 01:01:08.090 --> 01:01:10.900 align:middle line:90% So that's also in the algebra. 01:01:10.900 --> 01:01:14.170 align:middle line:84% And also, this implies that R, which 01:01:14.170 --> 01:01:18.310 align:middle line:84% is equal to the complement of the empty set, 01:01:18.310 --> 01:01:19.550 align:middle line:90% is also in the algebra. 01:01:19.550 --> 01:01:24.130 align:middle line:84% So for algebras of sets, they always 01:01:24.130 --> 01:01:26.680 align:middle line:84% contain the empty set for nonempty ones, 01:01:26.680 --> 01:01:29.260 align:middle line:84% which is the only kind we're ever going to care about. 01:01:29.260 --> 01:01:31.924 align:middle line:84% They always contain the empty set in R. 01:01:31.924 --> 01:01:37.150 align:middle line:84% And not only are they closed under taking complements, 01:01:37.150 --> 01:01:40.920 align:middle line:90% but also finite unions. 01:01:40.920 --> 01:01:46.560 align:middle line:84% And just like we proved that for algebras, finite intersections 01:01:46.560 --> 01:01:48.930 align:middle line:84% are also in the algebra, you can also 01:01:48.930 --> 01:01:53.070 align:middle line:84% prove that for a sigma algebra, countable intersections 01:01:53.070 --> 01:01:54.580 align:middle line:90% are also in the sigma algebra. 01:01:54.580 --> 01:01:56.340 align:middle line:90% So let me make a point of that. 01:01:56.340 --> 01:02:02.980 align:middle line:90% 01:02:02.980 --> 01:02:17.240 align:middle line:84% If A is a sigma algebra, then E n, a countable collection 01:02:17.240 --> 01:02:20.540 align:middle line:84% of elements of the sigma algebra, 01:02:20.540 --> 01:02:28.980 align:middle line:84% implies that this countable intersection is also 01:02:28.980 --> 01:02:32.160 align:middle line:84% in the algebra, again by De Morgan's law. 01:02:32.160 --> 01:02:40.840 align:middle line:90% 01:02:40.840 --> 01:02:44.440 align:middle line:84% So why am I making all these general definitions now? 01:02:44.440 --> 01:02:52.520 align:middle line:84% So what we're going to show soon, 01:02:52.520 --> 01:02:55.940 align:middle line:84% at least in the next lecture, we'll 01:02:55.940 --> 01:03:00.090 align:middle line:84% soon show that if I define script 01:03:00.090 --> 01:03:12.230 align:middle line:84% M as a set of all measurable subsets of R, 01:03:12.230 --> 01:03:16.490 align:middle line:84% so this is a collection of subsets of R, 01:03:16.490 --> 01:03:17.915 align:middle line:90% that this is a sigma algebra-- 01:03:17.915 --> 01:03:21.880 align:middle line:90% 01:03:21.880 --> 01:03:23.260 align:middle line:90% one of the things we'll show. 01:03:23.260 --> 01:03:28.440 align:middle line:90% 01:03:28.440 --> 01:03:33.850 align:middle line:84% Now how we're doing this now is we 01:03:33.850 --> 01:03:37.300 align:middle line:84% had these ideals of what a measure should be, 01:03:37.300 --> 01:03:43.700 align:middle line:84% and we're building it up from this way, and in the end 01:03:43.700 --> 01:03:47.540 align:middle line:84% we're going to come up with this collection of measurable sets, 01:03:47.540 --> 01:03:51.350 align:middle line:84% which will have this special structure of being a sigma 01:03:51.350 --> 01:03:52.370 align:middle line:90% algebra. 01:03:52.370 --> 01:03:53.870 align:middle line:84% And then our measure will be defined 01:03:53.870 --> 01:03:57.200 align:middle line:90% on this sigma algebra of sets. 01:03:57.200 --> 01:03:59.450 align:middle line:84% When you go on to general measure theory, 01:03:59.450 --> 01:04:02.660 align:middle line:90% that is the input for you. 01:04:02.660 --> 01:04:05.840 align:middle line:84% A measure space will then be a collection, 01:04:05.840 --> 01:04:10.730 align:middle line:84% a sigma algebra of subsets of a set, with a measure on it. 01:04:10.730 --> 01:04:11.660 align:middle line:90% That's your input. 01:04:11.660 --> 01:04:15.050 align:middle line:84% Here, your building, if you like, 01:04:15.050 --> 01:04:18.470 align:middle line:84% one of the first nontrivial measures. 01:04:18.470 --> 01:04:26.655 align:middle line:84% That's what we're doing, one of the most important measures, 01:04:26.655 --> 01:04:27.155 align:middle line:90% really. 01:04:27.155 --> 01:04:30.910 align:middle line:90% 01:04:30.910 --> 01:04:33.610 align:middle line:90% So we have that definition. 01:04:33.610 --> 01:04:37.700 align:middle line:84% If you've taken my class before, you 01:04:37.700 --> 01:04:41.180 align:middle line:84% know that if there's a definition, 01:04:41.180 --> 01:04:43.370 align:middle line:84% then we should see an example or two. 01:04:43.370 --> 01:04:48.200 align:middle line:84% I said that the set of measurable sets 01:04:48.200 --> 01:04:49.310 align:middle line:90% will be a sigma algebra. 01:04:49.310 --> 01:04:51.380 align:middle line:84% That's going to take a little work to get to, 01:04:51.380 --> 01:04:55.160 align:middle line:84% but we can come up with a few examples already. 01:04:55.160 --> 01:05:12.170 align:middle line:90% 01:05:12.170 --> 01:05:14.930 align:middle line:90% So some simple examples-- 01:05:14.930 --> 01:05:17.345 align:middle line:90% we have the stupidest example. 01:05:17.345 --> 01:05:19.220 align:middle line:84% Always start off with the stupidest examples. 01:05:19.220 --> 01:05:23.280 align:middle line:90% That's the way to go. 01:05:23.280 --> 01:05:32.240 align:middle line:84% So the simplest sigma algebra is given 01:05:32.240 --> 01:05:35.900 align:middle line:84% by this collection of subsets consisting only 01:05:35.900 --> 01:05:42.850 align:middle line:84% of the empty set in R. The next stupidest one 01:05:42.850 --> 01:05:47.170 align:middle line:84% is on the other end of the spectrum, where every subset is 01:05:47.170 --> 01:05:49.660 align:middle line:90% in this sigma algebra. 01:05:49.660 --> 01:05:56.690 align:middle line:90% So these are sigma algebras. 01:05:56.690 --> 01:06:01.680 align:middle line:90% 01:06:01.680 --> 01:06:04.890 align:middle line:90% What's a non-stupid one? 01:06:04.890 --> 01:06:08.790 align:middle line:84% Let's say we take A to be the set, the collection 01:06:08.790 --> 01:06:18.590 align:middle line:84% of all subsets E, such that either E is countable 01:06:18.590 --> 01:06:27.570 align:middle line:90% or E complement is countable. 01:06:27.570 --> 01:06:37.640 align:middle line:90% 01:06:37.640 --> 01:06:42.720 align:middle line:90% So why is this? 01:06:42.720 --> 01:06:44.550 align:middle line:84% I claim this is also a sigma algebra. 01:06:44.550 --> 01:06:54.140 align:middle line:90% 01:06:54.140 --> 01:06:58.100 align:middle line:90% Why is this a sigma algebra? 01:06:58.100 --> 01:07:03.650 align:middle line:84% First off, it's clear that if E is in A then its complement is 01:07:03.650 --> 01:07:10.210 align:middle line:84% in A. Because if E is countable, then 01:07:10.210 --> 01:07:13.780 align:middle line:84% the complement of its complement is E, is countable. 01:07:13.780 --> 01:07:17.080 align:middle line:84% So E in A implies that E complement 01:07:17.080 --> 01:07:20.860 align:middle line:84% is A. In other words, this condition is symmetric in E 01:07:20.860 --> 01:07:22.570 align:middle line:90% and E complement. 01:07:22.570 --> 01:07:29.070 align:middle line:84% Why is it closed under taking countable unions? 01:07:29.070 --> 01:07:42.410 align:middle line:84% So suppose I take a collection of elements from my collection 01:07:42.410 --> 01:07:47.420 align:middle line:84% here, A. I want to show the union is in A. 01:07:47.420 --> 01:08:02.390 align:middle line:84% Then if for all n, E n is countable, 01:08:02.390 --> 01:08:11.850 align:middle line:84% then the union over n of E sub n is a countable union 01:08:11.850 --> 01:08:13.170 align:middle line:90% of countable sets. 01:08:13.170 --> 01:08:17.430 align:middle line:84% And therefore, this is countable, 01:08:17.430 --> 01:08:31.220 align:middle line:84% which implies this guy is in the set collection A. 01:08:31.220 --> 01:08:33.439 align:middle line:84% So this is the case, that all of these 01:08:33.439 --> 01:08:39.430 align:middle line:84% are countable if there exists an integer N0 such 01:08:39.430 --> 01:08:52.899 align:middle line:84% that E n complement is countable, E n sub 0. 01:08:52.899 --> 01:08:59.630 align:middle line:84% Then, remember what I have to verify 01:08:59.630 --> 01:09:04.939 align:middle line:84% is that the union of E n's is in the set 01:09:04.939 --> 01:09:07.069 align:middle line:84% A, which means either it's countable 01:09:07.069 --> 01:09:09.660 align:middle line:90% or its complement is countable. 01:09:09.660 --> 01:09:15.490 align:middle line:84% So if I look at the complement of the union, 01:09:15.490 --> 01:09:19.300 align:middle line:84% this is then equal to, by De Morgan's law, 01:09:19.300 --> 01:09:21.220 align:middle line:84% the intersection of the complements. 01:09:21.220 --> 01:09:26.330 align:middle line:84% And this is contained in one of these guys. 01:09:26.330 --> 01:09:32.080 align:middle line:84% And therefore, this is a subset of a countable set, which 01:09:32.080 --> 01:09:41.670 align:middle line:84% implies that this is countable, and which 01:09:41.670 --> 01:09:55.290 align:middle line:84% is one of the conditions of A. And therefore, this is in A. 01:09:55.290 --> 01:09:58.770 align:middle line:84% So therefore, this collection of subsets 01:09:58.770 --> 01:10:02.760 align:middle line:84% of R such that E is countable or the complement of E 01:10:02.760 --> 01:10:07.680 align:middle line:84% is countable is a sigma algebra of sets. 01:10:07.680 --> 01:10:13.845 align:middle line:84% It's usually referred to as the co-countable sigma algebra. 01:10:13.845 --> 01:10:19.420 align:middle line:90% 01:10:19.420 --> 01:10:23.260 align:middle line:84% And one can define a measure on this collection, this sigma 01:10:23.260 --> 01:10:24.550 align:middle line:90% algebra. 01:10:24.550 --> 01:10:32.640 align:middle line:84% But we're only interested in Lebesgue measure, which 01:10:32.640 --> 01:10:34.170 align:middle line:84% is going to be defined on, as we'll 01:10:34.170 --> 01:10:39.510 align:middle line:84% see, the sigma algebra of measurable subsets of R, 01:10:39.510 --> 01:10:46.720 align:middle line:84% Lebesgue measurable subsets of R. 01:10:46.720 --> 01:10:53.725 align:middle line:90% So let's do one more example. 01:10:53.725 --> 01:10:58.837 align:middle line:90% 01:10:58.837 --> 01:11:01.170 align:middle line:84% This will just take a minute and it's not too technical. 01:11:01.170 --> 01:11:02.628 align:middle line:84% I know it's the end of the lecture. 01:11:02.628 --> 01:11:04.980 align:middle line:84% So maybe-- well, I mean, you're at home watching this. 01:11:04.980 --> 01:11:08.070 align:middle line:84% You can pause it at some point, get a snack, 01:11:08.070 --> 01:11:09.960 align:middle line:90% change pajamas, whatever. 01:11:09.960 --> 01:11:12.450 align:middle line:84% But anyways, here's one last example 01:11:12.450 --> 01:11:15.690 align:middle line:84% of a sigma algebra, which is, in fact, a very important sigma 01:11:15.690 --> 01:11:25.440 align:middle line:90% algebra, which is the following. 01:11:25.440 --> 01:11:28.310 align:middle line:90% 01:11:28.310 --> 01:11:33.220 align:middle line:84% Let capital sigma be the collection 01:11:33.220 --> 01:11:48.880 align:middle line:84% of all sigma algebras containing all open sets. 01:11:48.880 --> 01:11:59.040 align:middle line:90% 01:11:59.040 --> 01:12:02.730 align:middle line:84% So let's unwind this definition for a minute. 01:12:02.730 --> 01:12:05.540 align:middle line:84% So you are in this collection of sigma algebras 01:12:05.540 --> 01:12:08.540 align:middle line:84% if you are a sigma algebra and you contain all 01:12:08.540 --> 01:12:15.310 align:middle line:84% open subsets of R. So for example, the power set of R, 01:12:15.310 --> 01:12:18.330 align:middle line:84% which is a sigma algebra-- a trivial one-- contains 01:12:18.330 --> 01:12:20.250 align:middle line:84% every subset of R, so it certainly 01:12:20.250 --> 01:12:23.940 align:middle line:84% contains every open subset of R. This 01:12:23.940 --> 01:12:27.705 align:middle line:84% is in this collection of sigma algebras. 01:12:27.705 --> 01:12:37.630 align:middle line:90% 01:12:37.630 --> 01:12:46.390 align:middle line:84% And let me define script B to be the intersection over all sigma 01:12:46.390 --> 01:12:49.280 align:middle line:84% algebras in this collection of sigma algebras. 01:12:49.280 --> 01:12:53.650 align:middle line:84% So this is the intersection of every sigma algebra containing 01:12:53.650 --> 01:12:56.360 align:middle line:90% all open sets. 01:12:56.360 --> 01:13:00.910 align:middle line:90% So first off, this is nonempty. 01:13:00.910 --> 01:13:06.130 align:middle line:84% This is a nonempty intersection, so note, this 01:13:06.130 --> 01:13:10.540 align:middle line:84% is a intersection of subsets of collections of subsets of R 01:13:10.540 --> 01:13:16.750 align:middle line:90% So this is R and it's nonempty. 01:13:16.750 --> 01:13:22.270 align:middle line:90% 01:13:22.270 --> 01:13:25.380 align:middle line:90% Then B is-- 01:13:25.380 --> 01:13:27.390 align:middle line:84% I'm going to say a lot in these next few words-- 01:13:27.390 --> 01:13:41.330 align:middle line:84% is the smallest sigma algebra containing 01:13:41.330 --> 01:13:51.000 align:middle line:84% all open subsets of R. And we call it the Borel sigma 01:13:51.000 --> 01:13:51.500 align:middle line:90% algebra. 01:13:51.500 --> 01:14:08.710 align:middle line:90% 01:14:08.710 --> 01:14:10.300 align:middle line:90% So how do you think of this? 01:14:10.300 --> 01:14:15.820 align:middle line:84% You take every sigma algebra in the universe 01:14:15.820 --> 01:14:18.850 align:middle line:90% that contain every open set. 01:14:18.850 --> 01:14:21.520 align:middle line:84% You take the intersection of all these sigma algebras. 01:14:21.520 --> 01:14:24.020 align:middle line:84% My claim is that you get a sigma algebra, 01:14:24.020 --> 01:14:26.530 align:middle line:84% and this is the smallest sigma algebra containing 01:14:26.530 --> 01:14:30.390 align:middle line:90% all open subsets of R. 01:14:30.390 --> 01:14:33.090 align:middle line:84% So another way to say this last sentence 01:14:33.090 --> 01:14:48.570 align:middle line:84% is, E is in sigma and for all A in sigma, E is contained in A. 01:14:48.570 --> 01:14:51.630 align:middle line:84% So it's the smallest sigma algebra containing 01:14:51.630 --> 01:15:00.530 align:middle line:84% all open subsets of R. The proof is not hard. 01:15:00.530 --> 01:15:03.430 align:middle line:84% You just need to make sure you understand 01:15:03.430 --> 01:15:05.200 align:middle line:90% exactly what I'm saying here. 01:15:05.200 --> 01:15:12.060 align:middle line:84% And if you understand what I'm saying here, 01:15:12.060 --> 01:15:13.580 align:middle line:90% the proof is quite simple. 01:15:13.580 --> 01:15:16.580 align:middle line:84% And I'm only going to do one part. 01:15:16.580 --> 01:15:23.560 align:middle line:84% So first off, once I show that B is a sigma algebra, 01:15:23.560 --> 01:15:31.900 align:middle line:84% then the rest follows because if I take any open subset of R, 01:15:31.900 --> 01:15:34.540 align:middle line:84% it's contained in every one of these. 01:15:34.540 --> 01:15:36.640 align:middle line:84% And therefore, it's contained in the intersection. 01:15:36.640 --> 01:15:39.100 align:middle line:84% So every open subset is contained 01:15:39.100 --> 01:15:42.010 align:middle line:84% in B or every open subset is an element of B. 01:15:42.010 --> 01:15:47.170 align:middle line:84% I shouldn't say contained, but every open subset is 01:15:47.170 --> 01:15:53.210 align:middle line:84% an element of B. And so I just need to show 01:15:53.210 --> 01:15:54.710 align:middle line:90% that B is a sigma algebra. 01:15:54.710 --> 01:15:58.640 align:middle line:84% And since it's equal to the intersection of all sigma 01:15:58.640 --> 01:16:02.340 align:middle line:84% algebras containing every open subset, 01:16:02.340 --> 01:16:03.870 align:middle line:90% it has to be the smallest one. 01:16:03.870 --> 01:16:08.220 align:middle line:84% Any other one-- this intersection for any fixed one, 01:16:08.220 --> 01:16:11.190 align:middle line:84% this intersection is contained in any fixed one. 01:16:11.190 --> 01:16:15.150 align:middle line:84% And therefore, B is contained in any fixed sigma algebra which 01:16:15.150 --> 01:16:16.950 align:middle line:90% contains all the open subsets. 01:16:16.950 --> 01:16:22.140 align:middle line:84% So I just need to verify that B is a sigma algebra. 01:16:22.140 --> 01:16:31.940 align:middle line:90% 01:16:31.940 --> 01:16:37.930 align:middle line:84% And since I'm running out of time, I'm going to do one part, 01:16:37.930 --> 01:16:41.590 align:middle line:84% or verify just one part of the definition. 01:16:41.590 --> 01:16:45.900 align:middle line:84% Because the other part's essentially the same except you 01:16:45.900 --> 01:16:48.840 align:middle line:90% have to use more chalk. 01:16:48.840 --> 01:16:54.150 align:middle line:84% So I just need to verify that B is a sigma algebra. 01:16:54.150 --> 01:16:59.160 align:middle line:90% 01:16:59.160 --> 01:17:02.940 align:middle line:84% Now, suppose E is an element of B. 01:17:02.940 --> 01:17:07.140 align:middle line:84% So this means it's a subset of R. It's 01:17:07.140 --> 01:17:13.950 align:middle line:84% one of these subsets of R that's an element of B. 01:17:13.950 --> 01:17:20.910 align:middle line:84% Then for all sigma algebras in this collection, 01:17:20.910 --> 01:17:26.490 align:middle line:84% since B is the intersection over all of these sigma algebras, 01:17:26.490 --> 01:17:38.300 align:middle line:84% E is in A. And since each of these is a sigma algebra, 01:17:38.300 --> 01:17:41.771 align:middle line:84% I get that the complement is in A. 01:17:41.771 --> 01:17:45.800 align:middle line:84% And this statement says for every sigma 01:17:45.800 --> 01:17:47.870 align:middle line:84% algebra in this collection, the complement 01:17:47.870 --> 01:17:52.430 align:middle line:84% is in that sigma algebra, and therefore, E complement 01:17:52.430 --> 01:17:59.410 align:middle line:84% is in the intersection, which is, remember, 01:17:59.410 --> 01:18:02.200 align:middle line:84% how I've defined the Borel sigma algebra. 01:18:02.200 --> 01:18:06.430 align:middle line:90% 01:18:06.430 --> 01:18:11.530 align:middle line:84% And again, the proof of being closed under countable unions 01:18:11.530 --> 01:18:12.760 align:middle line:90% is sort of the same. 01:18:12.760 --> 01:18:17.290 align:middle line:84% Take a countable collection of elements of B. Then 01:18:17.290 --> 01:18:20.020 align:middle line:90% this countable collection-- 01:18:20.020 --> 01:18:23.770 align:middle line:84% they all have to be elements of A for every A in sigma. 01:18:23.770 --> 01:18:26.110 align:middle line:84% Therefore, their union has to be an element 01:18:26.110 --> 01:18:29.830 align:middle line:84% of A for every A in sigma because A is a sigma algebra. 01:18:29.830 --> 01:18:32.710 align:middle line:84% And therefore, the union is in the intersection 01:18:32.710 --> 01:18:37.255 align:middle line:84% over all A's which is equal to B, the Borel sigma algebra. 01:18:37.255 --> 01:18:40.600 align:middle line:90% 01:18:40.600 --> 01:18:43.450 align:middle line:84% So what we're working towards and what 01:18:43.450 --> 01:18:51.100 align:middle line:84% we're going to do next lecture is we're going to show that M, 01:18:51.100 --> 01:18:57.190 align:middle line:84% the collection of Lebesgue measurable subsets of R 01:18:57.190 --> 01:19:01.750 align:middle line:84% is a sigma algebra and it contains the Borel sigma 01:19:01.750 --> 01:19:02.750 align:middle line:90% algebra. 01:19:02.750 --> 01:19:09.760 align:middle line:84% So it contains this very big collection of subsets of R. 01:19:09.760 --> 01:19:12.820 align:middle line:84% I mean, this is truly a very big subset of R, like I was saying. 01:19:12.820 --> 01:19:14.740 align:middle line:90% It contains all open subsets. 01:19:14.740 --> 01:19:16.390 align:middle line:84% And since it's a sigma algebra, it 01:19:16.390 --> 01:19:18.760 align:middle line:84% contains the complements of all open subsets, i.e. 01:19:18.760 --> 01:19:20.350 align:middle line:90% closed subsets. 01:19:20.350 --> 01:19:24.550 align:middle line:84% but then it also contains all intersections of open subsets 01:19:24.550 --> 01:19:30.170 align:middle line:84% because sigma algebras are closed under taking 01:19:30.170 --> 01:19:32.730 align:middle line:84% countable intersections, as well. 01:19:32.730 --> 01:19:37.620 align:middle line:84% And then, you could take countable unions 01:19:37.620 --> 01:19:40.410 align:middle line:84% of those countable intersections of open subsets, 01:19:40.410 --> 01:19:43.300 align:middle line:90% and then, and so on, and so on. 01:19:43.300 --> 01:19:48.030 align:middle line:84% So you get a very rich class of subsets of R that's 01:19:48.030 --> 01:19:50.100 align:middle line:90% contained in the sigma algebra. 01:19:50.100 --> 01:19:54.320 align:middle line:84% And what we're going to show is that, like I said, 01:19:54.320 --> 01:19:57.890 align:middle line:84% the collection of Lebesgue measurable subsets 01:19:57.890 --> 01:20:00.980 align:middle line:84% of R, that's a sigma algebra and it contains the Borel sigma 01:20:00.980 --> 01:20:01.500 align:middle line:90% algebra. 01:20:01.500 --> 01:20:03.890 align:middle line:84% So it's a very rich class of subsets, 01:20:03.890 --> 01:20:06.380 align:middle line:84% even though up to this point, all we've shown 01:20:06.380 --> 01:20:10.280 align:middle line:84% is that the empty set R and the sets 01:20:10.280 --> 01:20:13.640 align:middle line:84% with outer measure equal to 0 are Lebesgue measurable. 01:20:13.640 --> 01:20:15.020 align:middle line:90% So we'll do that next time. 01:20:15.020 --> 01:20:17.650 align:middle line:90% And I'll stop there. 01:20:17.650 --> 01:20:24.000 align:middle line:90%