WEBVTT 00:00:00.000 --> 00:00:01.488 align:middle line:90% [SQUEAKING] 00:00:01.488 --> 00:00:03.472 align:middle line:90% [RUSTLING] 00:00:03.472 --> 00:00:05.952 align:middle line:90% [CLICKING] 00:00:05.952 --> 00:00:19.870 align:middle line:90% 00:00:19.870 --> 00:00:22.090 align:middle line:84% CASEY RODRIGUEZ: All right, so let's continue 00:00:22.090 --> 00:00:24.070 align:middle line:84% with our discussion of measurable functions. 00:00:24.070 --> 00:00:39.820 align:middle line:84% So last time we introduced the notion of measurable functions. 00:00:39.820 --> 00:00:59.170 align:middle line:84% So if I have a measurable set, E from-- 00:00:59.170 --> 00:01:01.600 align:middle line:84% or f from E to the extended real numbers 00:01:01.600 --> 00:01:07.410 align:middle line:84% is measurable, or is a measurable function, 00:01:07.410 --> 00:01:20.490 align:middle line:84% if for all alpha in R, the inverse image of the half 00:01:20.490 --> 00:01:25.200 align:middle line:84% infinite open interval alpha to infinity is measurable. 00:01:25.200 --> 00:01:33.670 align:middle line:90% 00:01:33.670 --> 00:01:37.120 align:middle line:84% And then we proved that if we have a function which 00:01:37.120 --> 00:01:41.230 align:middle line:84% is measurable, then not only does 00:01:41.230 --> 00:01:43.690 align:middle line:84% is the pre-image of these intervals measurable, 00:01:43.690 --> 00:01:49.090 align:middle line:84% but the pre-image of any Borel set, any member of the Borel 00:01:49.090 --> 00:01:51.460 align:middle line:84% sigma algebra, which includes open sets, close sets, 00:01:51.460 --> 00:01:54.940 align:middle line:84% and so on, the pre-image of those sets is also measurable. 00:01:54.940 --> 00:01:57.720 align:middle line:90% 00:01:57.720 --> 00:02:00.960 align:middle line:84% And we proved that being measurable 00:02:00.960 --> 00:02:06.300 align:middle line:84% is closed undertaking sups, imps, lim sups, lim imps, 00:02:06.300 --> 00:02:10.440 align:middle line:84% limits in particular, and changing 00:02:10.440 --> 00:02:12.540 align:middle line:90% the function a little bit on-- 00:02:12.540 --> 00:02:15.030 align:middle line:84% or changing a function on the set of measure 0 also 00:02:15.030 --> 00:02:17.580 align:middle line:90% preserves measurability. 00:02:17.580 --> 00:02:21.960 align:middle line:84% And also being measurable is closed 00:02:21.960 --> 00:02:26.700 align:middle line:84% under the algebraic operations of taking linear combinations 00:02:26.700 --> 00:02:27.450 align:middle line:90% and products. 00:02:27.450 --> 00:02:31.030 align:middle line:90% 00:02:31.030 --> 00:02:33.670 align:middle line:84% Now, this definition and properties we've worked out 00:02:33.670 --> 00:02:38.110 align:middle line:84% was for extended real-valued functions. 00:02:38.110 --> 00:02:42.720 align:middle line:84% Quite often we'll be dealing with-- 00:02:42.720 --> 00:02:45.540 align:middle line:84% or in general, functions that take 00:02:45.540 --> 00:02:47.020 align:middle line:90% values in the complex numbers. 00:02:47.020 --> 00:02:59.432 align:middle line:84% So let me define what it means for a complex value function 00:02:59.432 --> 00:03:00.140 align:middle line:90% to be measurable. 00:03:00.140 --> 00:03:02.300 align:middle line:90% It's not too crazy. 00:03:02.300 --> 00:03:06.580 align:middle line:84% Let E be a subset of R be measurable. 00:03:06.580 --> 00:03:15.440 align:middle line:84% We say a function f from E now to the complex numbers 00:03:15.440 --> 00:03:20.420 align:middle line:84% is measurable if the two functions given 00:03:20.420 --> 00:03:26.120 align:middle line:84% by the real part of f, which now goes from E to R 00:03:26.120 --> 00:03:32.480 align:middle line:84% and the imaginary part of f, which now is a function from E 00:03:32.480 --> 00:03:42.220 align:middle line:90% to R, are measurable. 00:03:42.220 --> 00:03:44.930 align:middle line:90% 00:03:44.930 --> 00:03:47.840 align:middle line:84% So function into the complex numbers 00:03:47.840 --> 00:03:56.630 align:middle line:84% you can always write as f equals the real part of f 00:03:56.630 --> 00:04:00.410 align:middle line:84% plus the imaginary part of f times i. 00:04:00.410 --> 00:04:04.070 align:middle line:84% So we just-- to say that f is measurable, 00:04:04.070 --> 00:04:09.360 align:middle line:84% just require that it's real and imaginary parts are measurable, 00:04:09.360 --> 00:04:09.860 align:middle line:90% OK. 00:04:09.860 --> 00:04:16.640 align:middle line:90% 00:04:16.640 --> 00:04:19.700 align:middle line:84% And then you can verify the following simple theorem. 00:04:19.700 --> 00:04:23.570 align:middle line:84% Maybe I'll put it on assignment or not, haven't decided yet-- 00:04:23.570 --> 00:04:27.050 align:middle line:90% or just parts of it. 00:04:27.050 --> 00:04:36.570 align:middle line:84% If f G were measurable-- and I didn't say 00:04:36.570 --> 00:04:39.360 align:middle line:84% so, but if we're talking about measurable functions , 00:04:39.360 --> 00:04:42.270 align:middle line:84% the domain E, the domain is always assumed to be 00:04:42.270 --> 00:04:44.670 align:middle line:90% a measurable subset of R. 00:04:44.670 --> 00:04:54.070 align:middle line:84% So if these are measurable functions and alpha is 00:04:54.070 --> 00:05:00.810 align:middle line:84% a complex number, then the function's 00:05:00.810 --> 00:05:07.218 align:middle line:84% alpha times f, f plus g, f times g-- 00:05:07.218 --> 00:05:08.760 align:middle line:84% and then we can do a few other things 00:05:08.760 --> 00:05:12.900 align:middle line:84% to complex numbers that change it. 00:05:12.900 --> 00:05:22.440 align:middle line:84% The complex conjugate of f and modulus of f 00:05:22.440 --> 00:05:27.360 align:middle line:90% are measurable functions. 00:05:27.360 --> 00:05:37.080 align:middle line:90% 00:05:37.080 --> 00:05:40.635 align:middle line:84% And then we have the following theorem. 00:05:40.635 --> 00:05:45.400 align:middle line:90% 00:05:45.400 --> 00:05:55.950 align:middle line:84% If fn from E to C is measurable for all n. 00:05:55.950 --> 00:06:21.310 align:middle line:84% And these functions converge pointwise to a function f, 00:06:21.310 --> 00:06:27.320 align:middle line:84% then the limiting function is measurable, OK. 00:06:27.320 --> 00:06:30.990 align:middle line:90% 00:06:30.990 --> 00:06:33.400 align:middle line:90% And this follows, again-- 00:06:33.400 --> 00:06:35.550 align:middle line:84% so both of these theorems follow immediately 00:06:35.550 --> 00:06:39.450 align:middle line:84% from what we know about measurable 00:06:39.450 --> 00:06:43.330 align:middle line:90% extended real-value functions. 00:06:43.330 --> 00:06:52.000 align:middle line:84% So, for example, this one here follows from the fact 00:06:52.000 --> 00:06:56.720 align:middle line:84% that limit n goes to infinity of fn 00:06:56.720 --> 00:07:09.930 align:middle line:84% of x equals f of x if and only if limit as n 00:07:09.930 --> 00:07:14.240 align:middle line:84% goes to infinity of the real part of f of f, 00:07:14.240 --> 00:07:23.830 align:middle line:84% fn of x equals the real part of f of x, and the limit as n 00:07:23.830 --> 00:07:26.620 align:middle line:84% goes to infinity of the imaginary part 00:07:26.620 --> 00:07:33.850 align:middle line:84% fn of f of x equals the imaginary part of f of x. 00:07:33.850 --> 00:07:36.790 align:middle line:84% And if we're assuming fn is measurable for all n, 00:07:36.790 --> 00:07:40.150 align:middle line:84% then the real part and imaginary parts of fn 00:07:40.150 --> 00:07:42.580 align:middle line:90% are measurable for all n. 00:07:42.580 --> 00:07:44.530 align:middle line:84% And therefore there are pointwise limits, 00:07:44.530 --> 00:07:47.290 align:middle line:84% which is the real part of f and imaginary part of f 00:07:47.290 --> 00:07:50.980 align:middle line:84% are measurable by the theorem we proved 00:07:50.980 --> 00:07:56.480 align:middle line:84% about extended real-value measurable functions. 00:07:56.480 --> 00:07:58.960 align:middle line:84% And so we conclude that the real part 00:07:58.960 --> 00:08:01.120 align:middle line:84% of f and the imaginary part are measurable, 00:08:01.120 --> 00:08:05.270 align:middle line:90% and therefore f is measurable. 00:08:05.270 --> 00:08:07.980 align:middle line:84% So you don't have to work very hard. 00:08:07.980 --> 00:08:10.940 align:middle line:84% You can just use of what we proved 00:08:10.940 --> 00:08:15.470 align:middle line:84% from the previous lecture about extended real-valued functions 00:08:15.470 --> 00:08:16.744 align:middle line:90% which are measurable. 00:08:16.744 --> 00:08:32.070 align:middle line:90% 00:08:32.070 --> 00:08:38.820 align:middle line:84% All right, so now as far as measurable functions go, 00:08:38.820 --> 00:08:47.100 align:middle line:84% we've shown that if I have a continuous function, then that 00:08:47.100 --> 00:08:48.990 align:middle line:84% is-- so this is from the previous time, 00:08:48.990 --> 00:08:51.660 align:middle line:84% from previous lecture, continuous functions 00:08:51.660 --> 00:08:52.350 align:middle line:90% are measurable. 00:08:52.350 --> 00:08:58.530 align:middle line:84% And if I have a measurable subset of E, 00:08:58.530 --> 00:09:01.930 align:middle line:84% then the indicator function of that set is measurable. 00:09:01.930 --> 00:09:04.305 align:middle line:84% And we know that linear combinations 00:09:04.305 --> 00:09:05.930 align:middle line:84% of measurable functions are measurable. 00:09:05.930 --> 00:09:11.070 align:middle line:84% So if I take linear combinations of indicator functions 00:09:11.070 --> 00:09:15.060 align:middle line:84% with, say, complex coefficients, then that 00:09:15.060 --> 00:09:18.150 align:middle line:90% will remain measurable as well. 00:09:18.150 --> 00:09:22.440 align:middle line:84% And those functions are the simplest type in that they only 00:09:22.440 --> 00:09:24.900 align:middle line:90% take finitely many values. 00:09:24.900 --> 00:09:29.790 align:middle line:84% They're so simple that we glorify them 00:09:29.790 --> 00:09:31.035 align:middle line:90% by giving them that name. 00:09:31.035 --> 00:09:33.890 align:middle line:90% 00:09:33.890 --> 00:09:38.760 align:middle line:84% And we'll show that every measurable function 00:09:38.760 --> 00:09:42.430 align:middle line:84% is, in a sense, approximately a simple function. 00:09:42.430 --> 00:09:54.780 align:middle line:84% So we have the following definition, if E is measurable, 00:09:54.780 --> 00:10:06.940 align:middle line:84% a measurable function phi from E to C is simple, 00:10:06.940 --> 00:10:10.090 align:middle line:90% or we call it a simple function. 00:10:10.090 --> 00:10:17.050 align:middle line:84% So we'll say is a simple function 00:10:17.050 --> 00:10:30.295 align:middle line:84% if the range of E phi of E is equal to finitely many values. 00:10:30.295 --> 00:10:36.650 align:middle line:90% 00:10:36.650 --> 00:10:39.950 align:middle line:84% So a measurable function is a simple function 00:10:39.950 --> 00:10:42.980 align:middle line:90% if its range is finite. 00:10:42.980 --> 00:10:48.130 align:middle line:84% So let me make a general remark about simple functions. 00:10:48.130 --> 00:10:51.430 align:middle line:84% And also when I write that phi of E is equal to a1 up to a n, 00:10:51.430 --> 00:10:54.230 align:middle line:90% this is a set. 00:10:54.230 --> 00:10:57.190 align:middle line:84% And when I write a1, a2, up to a n, 00:10:57.190 --> 00:10:59.650 align:middle line:90% I am implicitly writing here. 00:10:59.650 --> 00:11:01.000 align:middle line:90% I'm not saying the set-- 00:11:01.000 --> 00:11:02.680 align:middle line:84% I'm saying each of these elements 00:11:02.680 --> 00:11:04.870 align:middle line:90% are different from each other. 00:11:04.870 --> 00:11:08.050 align:middle line:84% So a simple function that just takes the value 1, 00:11:08.050 --> 00:11:11.930 align:middle line:84% I would not write its range as 1, 1, 1, 1. 00:11:11.930 --> 00:11:17.050 align:middle line:84% So although that's a kind of simple and silly remark 00:11:17.050 --> 00:11:19.240 align:middle line:84% to make, I just want to make it now. 00:11:19.240 --> 00:11:24.690 align:middle line:90% 00:11:24.690 --> 00:11:33.530 align:middle line:84% So if phi is a simple function, then we 00:11:33.530 --> 00:11:42.860 align:middle line:84% can write it in a canonical way with-- 00:11:42.860 --> 00:11:45.830 align:middle line:84% so if phi is a simple function with phi of E 00:11:45.830 --> 00:12:03.860 align:middle line:84% equals a1 up to a n, then for all i, the set a sub i, 00:12:03.860 --> 00:12:10.040 align:middle line:84% which is equal to the inverse image of-- 00:12:10.040 --> 00:12:15.960 align:middle line:84% so here phi is going from E to C. 00:12:15.960 --> 00:12:23.490 align:middle line:84% The inverse image of the single set, singleton, 00:12:23.490 --> 00:12:28.210 align:middle line:90% which is a closed-- 00:12:28.210 --> 00:12:31.315 align:middle line:90% well, this is going to be-- 00:12:31.315 --> 00:12:35.010 align:middle line:90% 00:12:35.010 --> 00:12:39.900 align:middle line:84% so the inverse image of this guy here 00:12:39.900 --> 00:12:50.830 align:middle line:84% is measurable because it's equal to the intersection of two 00:12:50.830 --> 00:12:51.790 align:middle line:90% measurable sets. 00:12:51.790 --> 00:12:56.860 align:middle line:84% It's equal to the intersection of when the real part of phi 00:12:56.860 --> 00:13:01.900 align:middle line:84% equals the real part of a sub i intersect the set of all x's 00:13:01.900 --> 00:13:03.880 align:middle line:84% where the imaginary part of phi equals 00:13:03.880 --> 00:13:05.860 align:middle line:90% the imaginary part of a sub i. 00:13:05.860 --> 00:13:14.340 align:middle line:84% So that's why the set is measurable, is measurable. 00:13:14.340 --> 00:13:19.185 align:middle line:84% And we have a few properties of this guy. 00:13:19.185 --> 00:13:24.450 align:middle line:90% 00:13:24.450 --> 00:13:28.540 align:middle line:84% If I take two different elements in the range-- 00:13:28.540 --> 00:13:30.450 align:middle line:84% again, this is why I made that comment 00:13:30.450 --> 00:13:33.060 align:middle line:84% that when I write the range this way, 00:13:33.060 --> 00:13:37.530 align:middle line:84% I'm listing the distinct elements of the range-- 00:13:37.530 --> 00:13:39.000 align:middle line:90% or the image of E-- 00:13:39.000 --> 00:13:46.720 align:middle line:84% For all i equals not equal to j, these two sets are disjoint. 00:13:46.720 --> 00:13:51.170 align:middle line:90% 00:13:51.170 --> 00:13:55.970 align:middle line:84% And if I take the union of i equals 1 to n 00:13:55.970 --> 00:14:00.570 align:middle line:84% of the ai's, this is equal to the total set E 00:14:00.570 --> 00:14:02.850 align:middle line:84% because this is just the inverse image 00:14:02.850 --> 00:14:09.500 align:middle line:84% of this set, which is just E, since E of E equals that, OK. 00:14:09.500 --> 00:14:12.280 align:middle line:90% 00:14:12.280 --> 00:14:23.060 align:middle line:84% And finally for all x in E, I can 00:14:23.060 --> 00:14:29.700 align:middle line:84% write phi of x as sum from i equals 1 to n 00:14:29.700 --> 00:14:37.180 align:middle line:90% of ai chi of a sub i of x. 00:14:37.180 --> 00:14:41.290 align:middle line:84% So for a simple function, I can write it in a canonical way 00:14:41.290 --> 00:14:46.270 align:middle line:84% where it's just a linear combination of indicator 00:14:46.270 --> 00:14:49.240 align:middle line:84% functions where the sets those indicator functions 00:14:49.240 --> 00:14:53.020 align:middle line:84% are non-zero on or disjoint from each other. 00:14:53.020 --> 00:15:00.245 align:middle line:84% And their union gives me E, the domain, OK. 00:15:00.245 --> 00:15:08.030 align:middle line:90% 00:15:08.030 --> 00:15:13.702 align:middle line:84% So these three are the simple but important properties 00:15:13.702 --> 00:15:15.285 align:middle line:84% of how to represent a simple function. 00:15:15.285 --> 00:15:19.450 align:middle line:90% 00:15:19.450 --> 00:15:22.750 align:middle line:84% So it's not difficult to verify, again, 00:15:22.750 --> 00:15:25.710 align:middle line:84% just from the definition of a simple function, 00:15:25.710 --> 00:15:53.920 align:middle line:84% that scalar multiples, linear combinations, and products 00:15:53.920 --> 00:16:05.130 align:middle line:84% of simple functions are, again, simple functions. 00:16:05.130 --> 00:16:23.240 align:middle line:90% 00:16:23.240 --> 00:16:26.240 align:middle line:84% So I've said that these functions are 00:16:26.240 --> 00:16:29.420 align:middle line:84% so simple that that's what we call them, 00:16:29.420 --> 00:16:36.980 align:middle line:84% and that they are somehow universal that, in a sense, 00:16:36.980 --> 00:16:40.445 align:middle line:84% any measurable function is almost a simple function. 00:16:40.445 --> 00:16:44.340 align:middle line:90% 00:16:44.340 --> 00:16:46.180 align:middle line:90% So in what sense do I mean that? 00:16:46.180 --> 00:16:50.920 align:middle line:84% So that's the content of the following theorem, 00:16:50.920 --> 00:16:59.140 align:middle line:90% which is that if f from E to-- 00:16:59.140 --> 00:17:02.770 align:middle line:84% and let's first-- we're going to do 00:17:02.770 --> 00:17:05.410 align:middle line:84% this for extended real-valued functions. 00:17:05.410 --> 00:17:08.859 align:middle line:84% And then the proof will essentially carry over 00:17:08.859 --> 00:17:09.980 align:middle line:90% to the complex value. 00:17:09.980 --> 00:17:13.390 align:middle line:84% So I'm just going to do it for the extended real-value 00:17:13.390 --> 00:17:16.869 align:middle line:90% non-negative functions. 00:17:16.869 --> 00:17:20.140 align:middle line:84% And, again, I'll indicate what the difference 00:17:20.140 --> 00:17:23.690 align:middle line:84% is once we go to complex valued measurable functions. 00:17:23.690 --> 00:17:29.070 align:middle line:90% So if E is from-- 00:17:29.070 --> 00:17:32.870 align:middle line:84% so now this is an extended real value, but it's non-negative. 00:17:32.870 --> 00:17:37.710 align:middle line:84% So if this is a measurable function, 00:17:37.710 --> 00:17:52.290 align:middle line:84% then there exists a sequence of simple functions 00:17:52.290 --> 00:18:00.630 align:middle line:84% phi sub n such that three things hold. 00:18:00.630 --> 00:18:02.670 align:middle line:90% We can do this-- 00:18:02.670 --> 00:18:05.100 align:middle line:84% let's write down-- let's go over here. 00:18:05.100 --> 00:18:13.860 align:middle line:90% 00:18:13.860 --> 00:18:16.180 align:middle line:90% three things hold. 00:18:16.180 --> 00:18:18.135 align:middle line:90% For all x in E-- 00:18:18.135 --> 00:18:23.670 align:middle line:90% 00:18:23.670 --> 00:18:28.140 align:middle line:84% or let me hold off on stating that. 00:18:28.140 --> 00:18:30.330 align:middle line:84% F dominates these simple functions. 00:18:30.330 --> 00:18:34.200 align:middle line:84% And these simple functions are pointwise increasing. 00:18:34.200 --> 00:18:39.336 align:middle line:84% For all x in E, 0 is less than or equal to phi 1 of-- 00:18:39.336 --> 00:18:43.480 align:middle line:84% phi 0 of x is less than or equal to phi 1 of x is less than 00:18:43.480 --> 00:18:48.860 align:middle line:84% or equal to phi 2 of x and so on. 00:18:48.860 --> 00:18:54.380 align:middle line:90% And they all sit below f of x. 00:18:54.380 --> 00:19:08.480 align:middle line:84% For all x in E, these phi's are converging to f of x. 00:19:08.480 --> 00:19:12.170 align:middle line:84% So A and B says that there exists 00:19:12.170 --> 00:19:15.080 align:middle line:84% a sequence, a measure of simple functions 00:19:15.080 --> 00:19:18.290 align:middle line:90% that increase to f of x. 00:19:18.290 --> 00:19:20.735 align:middle line:84% The last part is that if this function, 00:19:20.735 --> 00:19:24.470 align:middle line:84% if this measurable function is bounded, 00:19:24.470 --> 00:19:27.440 align:middle line:84% or wherever it's bounded, this convergence is not just 00:19:27.440 --> 00:19:28.475 align:middle line:90% pointwise but uniform. 00:19:28.475 --> 00:19:31.360 align:middle line:90% 00:19:31.360 --> 00:19:35.380 align:middle line:84% For all B bigger than or equal to 0, 00:19:35.380 --> 00:19:44.500 align:middle line:84% the sequence phi n converges to f uniformly on the set 00:19:44.500 --> 00:19:45.685 align:middle line:90% where f is bounded. 00:19:45.685 --> 00:19:56.150 align:middle line:90% 00:19:56.150 --> 00:20:01.490 align:middle line:84% So, again, the take home is that for every non-negative extended 00:20:01.490 --> 00:20:03.650 align:middle line:84% real-value measurable function, we 00:20:03.650 --> 00:20:06.050 align:middle line:84% can find a sequence of simple functions 00:20:06.050 --> 00:20:09.470 align:middle line:90% that well approximates f. 00:20:09.470 --> 00:20:12.230 align:middle line:84% And in what sense does this will approximate f, 00:20:12.230 --> 00:20:13.970 align:middle line:90% they increase to f. 00:20:13.970 --> 00:20:18.440 align:middle line:84% And if f is bounded, then that convergence is, in fact, 00:20:18.440 --> 00:20:19.070 align:middle line:90% uniform. 00:20:19.070 --> 00:20:22.910 align:middle line:84% Or more precisely, wherever f is bounded, 00:20:22.910 --> 00:20:25.980 align:middle line:90% convergence is uniform. 00:20:25.980 --> 00:20:30.860 align:middle line:84% So this is in what sense every measurable function is 00:20:30.860 --> 00:20:32.630 align:middle line:90% almost a simple function. 00:20:32.630 --> 00:20:40.540 align:middle line:90% 00:20:40.540 --> 00:20:47.850 align:middle line:84% So let me-- let's get started with the proof. 00:20:47.850 --> 00:20:50.660 align:middle line:84% And first I'm going to draw a few pictures so 00:20:50.660 --> 00:20:54.905 align:middle line:84% that you can get the idea of how we're going to do this, 00:20:54.905 --> 00:20:56.780 align:middle line:84% or how we're going to construct this sequence 00:20:56.780 --> 00:21:05.990 align:middle line:84% of simple functions, and then turn that into math which might 00:21:05.990 --> 00:21:09.290 align:middle line:84% be a little bit jarring if I went that route first and then 00:21:09.290 --> 00:21:12.073 align:middle line:90% drew pictures. 00:21:12.073 --> 00:21:13.490 align:middle line:84% And this picture I'm going to draw 00:21:13.490 --> 00:21:18.440 align:middle line:84% kind of looks like what I was talking about where we split 00:21:18.440 --> 00:21:22.430 align:middle line:84% the range up rather than the domain when we were motivating 00:21:22.430 --> 00:21:25.250 align:middle line:84% why we would even introduce the concept 00:21:25.250 --> 00:21:26.670 align:middle line:90% of a measurable function. 00:21:26.670 --> 00:21:32.700 align:middle line:84% So let's say we have our function f. 00:21:32.700 --> 00:21:36.600 align:middle line:90% 00:21:36.600 --> 00:21:41.820 align:middle line:84% Now how I'm going to build these phi's are that-- 00:21:41.820 --> 00:21:43.320 align:middle line:90% what I'm going to do is-- 00:21:43.320 --> 00:21:47.310 align:middle line:84% so here I'm going to draw phi 0 now. 00:21:47.310 --> 00:21:48.370 align:middle line:90% Where's the yellow chalk. 00:21:48.370 --> 00:21:53.440 align:middle line:90% 00:21:53.440 --> 00:21:58.300 align:middle line:84% What I do is this will indicate the power of 2, 00:21:58.300 --> 00:22:02.770 align:middle line:84% how high up I'm going, and also the resolution, how 00:22:02.770 --> 00:22:08.120 align:middle line:84% much of my dividing how high up I'm going into smaller parts. 00:22:08.120 --> 00:22:12.670 align:middle line:84% So phi 0 you should think that my height will be-- well, 00:22:12.670 --> 00:22:17.090 align:middle line:90% my height is going to be 1. 00:22:17.090 --> 00:22:24.550 align:middle line:84% And so what I do is, how I define this simple function 00:22:24.550 --> 00:22:31.870 align:middle line:84% phi 0 is I look at where f is above the final value 1. 00:22:31.870 --> 00:22:36.490 align:middle line:84% And my simple function will be that final value 1 there. 00:22:36.490 --> 00:22:43.910 align:middle line:90% 00:22:43.910 --> 00:22:47.530 align:middle line:90% And move this over a little bit. 00:22:47.530 --> 00:22:52.640 align:middle line:90% 00:22:52.640 --> 00:23:01.210 align:middle line:84% And that just leaves where f is less than 1. 00:23:01.210 --> 00:23:10.170 align:middle line:84% And there I set the value of my simple function to be 0, 00:23:10.170 --> 00:23:15.090 align:middle line:84% and 0 simply because that's the lower bound, 00:23:15.090 --> 00:23:20.790 align:middle line:84% or that's the smallest value of this interval 0, 1. 00:23:20.790 --> 00:23:24.060 align:middle line:90% And that's how I define phi 0. 00:23:24.060 --> 00:23:25.680 align:middle line:90% It goes up to height 1. 00:23:25.680 --> 00:23:33.600 align:middle line:84% And I only split each part into 1. 00:23:33.600 --> 00:23:35.250 align:middle line:90% So there's only one part here. 00:23:35.250 --> 00:23:38.520 align:middle line:84% Now maybe some of that didn't make any sense to you. 00:23:38.520 --> 00:23:39.570 align:middle line:90% That's OK. 00:23:39.570 --> 00:23:41.070 align:middle line:84% We're going to phi 1 and then I'm 00:23:41.070 --> 00:23:45.780 align:middle line:84% going to stop there because this is going to grow exponentially, 00:23:45.780 --> 00:23:49.500 align:middle line:84% which means I'll probably draw an exponentially 00:23:49.500 --> 00:23:53.080 align:middle line:90% worse picture each time. 00:23:53.080 --> 00:23:55.500 align:middle line:84% So now we're going to draw phi 2. 00:23:55.500 --> 00:23:57.240 align:middle line:90% And what I do is-- 00:23:57.240 --> 00:24:02.040 align:middle line:90% there's one-- or phi 1, sorry. 00:24:02.040 --> 00:24:05.100 align:middle line:84% One, again, should indicate the power of 2 00:24:05.100 --> 00:24:09.670 align:middle line:84% that I'm both going up and resolving the axis. 00:24:09.670 --> 00:24:13.140 align:middle line:90% So now I go up to-- 00:24:13.140 --> 00:24:16.570 align:middle line:90% let's make this a-- 00:24:16.570 --> 00:24:20.220 align:middle line:84% so, again, this will not be quite the scale, 00:24:20.220 --> 00:24:24.157 align:middle line:90% but hopefully that's OK. 00:24:24.157 --> 00:24:27.900 align:middle line:90% 00:24:27.900 --> 00:24:30.045 align:middle line:84% So let's make this a little closer to scale. 00:24:30.045 --> 00:24:34.150 align:middle line:90% 00:24:34.150 --> 00:24:39.790 align:middle line:84% OK, so one, now I have two parts, 2 to the 1. 00:24:39.790 --> 00:24:45.330 align:middle line:90% 00:24:45.330 --> 00:24:49.920 align:middle line:90% So this is parameterized how-- 00:24:49.920 --> 00:24:56.150 align:middle line:84% tallest I'm going and cutting up the range of my function f. 00:24:56.150 --> 00:25:05.210 align:middle line:84% And then I'm going to now resolve each part in 1/2, 2 00:25:05.210 --> 00:25:06.380 align:middle line:90% to the minus 1. 00:25:06.380 --> 00:25:07.700 align:middle line:90% So now I take-- 00:25:07.700 --> 00:25:10.950 align:middle line:90% 00:25:10.950 --> 00:25:18.840 align:middle line:84% this is now 3/2, and this is now 1/2, 2 to the minus 1. 00:25:18.840 --> 00:25:21.480 align:middle line:84% So why am I writing to 2 to the minus 1? 00:25:21.480 --> 00:25:26.980 align:middle line:84% Again because this one here corresponds to-- 00:25:26.980 --> 00:25:28.840 align:middle line:84% and that one here corresponds to-- 00:25:28.840 --> 00:25:33.700 align:middle line:84% I'm cutting the whole increments into halves. 00:25:33.700 --> 00:25:38.230 align:middle line:84% If I go on to phi 2, I'm going to be going up to 2 00:25:38.230 --> 00:25:43.780 align:middle line:84% to the 2, which is now 4; and from 4 to 3; 3 to 2; 2 to 1; 00:25:43.780 --> 00:25:47.860 align:middle line:84% I'm going to be cutting those into fourths. 00:25:47.860 --> 00:25:51.490 align:middle line:84% And, again, what I do is I look at-- 00:25:51.490 --> 00:26:10.660 align:middle line:84% I cut-- and now I look at where the function is 00:26:10.660 --> 00:26:11.710 align:middle line:90% in these widths. 00:26:11.710 --> 00:26:14.320 align:middle line:84% So if it's above my highest bound, 00:26:14.320 --> 00:26:20.500 align:middle line:84% then my simple function will be 2 on that set of x's where 00:26:20.500 --> 00:26:25.970 align:middle line:84% it's at its highest, or where f goes 00:26:25.970 --> 00:26:30.380 align:middle line:84% past the largest number which I'm resolving the axis in. 00:26:30.380 --> 00:26:33.400 align:middle line:90% 00:26:33.400 --> 00:26:35.365 align:middle line:90% And then now here, for example-- 00:26:35.365 --> 00:26:42.700 align:middle line:90% 00:26:42.700 --> 00:26:45.420 align:middle line:84% so you should try and draw your own picture on not just 00:26:45.420 --> 00:26:48.600 align:middle line:84% go off mine because, again, mine is looking kind of rough 00:26:48.600 --> 00:26:49.230 align:middle line:90% already. 00:26:49.230 --> 00:26:52.980 align:middle line:84% But now I look at the function-- the set 00:26:52.980 --> 00:26:57.060 align:middle line:84% of X's where the function is between 2 and 3/2. 00:26:57.060 --> 00:27:01.320 align:middle line:84% So that's going to be this piece and this piece. 00:27:01.320 --> 00:27:03.210 align:middle line:84% And then I set my simple function 00:27:03.210 --> 00:27:09.240 align:middle line:84% to be equal to the value, the lower bound on this interval 00:27:09.240 --> 00:27:10.560 align:middle line:90% that I've cut the range into. 00:27:10.560 --> 00:27:15.010 align:middle line:90% 00:27:15.010 --> 00:27:17.230 align:middle line:84% And then I do that from 3/2 to 1. 00:27:17.230 --> 00:27:20.440 align:middle line:84% And, for example, that's happening here on this x. 00:27:20.440 --> 00:27:21.850 align:middle line:90% And I set it equal to-- 00:27:21.850 --> 00:27:25.450 align:middle line:84% I set my simple function equal to, again, the lower bound 00:27:25.450 --> 00:27:29.890 align:middle line:84% on this interval that I've cut the range up into, which is 1, 00:27:29.890 --> 00:27:34.780 align:middle line:90% and so, and so on. 00:27:34.780 --> 00:27:40.030 align:middle line:90% 00:27:40.030 --> 00:27:43.048 align:middle line:84% And now the function between 1 and 1/2, so this 00:27:43.048 --> 00:27:44.340 align:middle line:90% is kind of only the last piece. 00:27:44.340 --> 00:27:45.930 align:middle line:90% Everywhere else is filled in. 00:27:45.930 --> 00:27:48.550 align:middle line:84% And there's no f in between here, 00:27:48.550 --> 00:27:52.140 align:middle line:90% so I don't assign a value there. 00:27:52.140 --> 00:27:55.970 align:middle line:90% 00:27:55.970 --> 00:27:58.540 align:middle line:84% All right, now that was me talking my way 00:27:58.540 --> 00:28:01.405 align:middle line:84% through how this function how the sequence of functions look. 00:28:01.405 --> 00:28:04.000 align:middle line:90% 00:28:04.000 --> 00:28:08.560 align:middle line:84% Now I'll just write down how these functions are defined. 00:28:08.560 --> 00:28:11.400 align:middle line:90% 00:28:11.400 --> 00:28:29.500 align:middle line:84% So for n equals 0, 1, 2, and so on for k between 2 00:28:29.500 --> 00:28:44.600 align:middle line:84% to the 2n minus 1, I define sets E kn. 00:28:44.600 --> 00:28:49.850 align:middle line:84% This is equal to the set of all X's in E such 00:28:49.850 --> 00:29:04.640 align:middle line:84% that f of x is between k times 2 to the minus n 00:29:04.640 --> 00:29:06.650 align:middle line:84% is less than or equal to k plus 1 2 00:29:06.650 --> 00:29:18.300 align:middle line:84% to the minus n, which is just me explicitly writing out 00:29:18.300 --> 00:29:21.510 align:middle line:84% that this is the inverse image of k times 00:29:21.510 --> 00:29:28.280 align:middle line:84% 2 to the minus n k plus 1 2 to the minus n closed. 00:29:28.280 --> 00:29:30.270 align:middle line:90% And this is an interval. 00:29:30.270 --> 00:29:33.200 align:middle line:84% And since f is measurable, assumed to be measurable, 00:29:33.200 --> 00:29:36.290 align:middle line:84% this is a measurable-- the inverse image of that interval 00:29:36.290 --> 00:29:41.110 align:middle line:90% is a measurable set, OK. 00:29:41.110 --> 00:29:55.170 align:middle line:84% And then I define fn to be the inverse image of when f exceeds 00:29:55.170 --> 00:30:02.250 align:middle line:84% my top value of how I'm cutting up the range, 2 the n, again, 00:30:02.250 --> 00:30:03.270 align:middle line:90% which is measurable. 00:30:03.270 --> 00:30:06.530 align:middle line:90% 00:30:06.530 --> 00:30:11.510 align:middle line:84% And, finally, I will take my simple function phi n 00:30:11.510 --> 00:30:22.010 align:middle line:84% to be sum from k equals 0 to 2 to the 2n minus 1, k times 2 00:30:22.010 --> 00:30:24.540 align:middle line:90% to the minus n, which is-- 00:30:24.540 --> 00:30:26.030 align:middle line:90% again, this is the lower-- 00:30:26.030 --> 00:30:30.290 align:middle line:84% this would correspond to the lower part of my E kn 00:30:30.290 --> 00:30:31.170 align:middle line:90% that I'm looking at. 00:30:31.170 --> 00:30:36.110 align:middle line:84% So for this example, if this Ekn is the two 3/2, 00:30:36.110 --> 00:30:57.810 align:middle line:84% and 3/2 is the lower part times chi E kn plus 2 to the n times 00:30:57.810 --> 00:31:01.560 align:middle line:90% where f exceeds 2 to the n. 00:31:01.560 --> 00:31:08.970 align:middle line:84% So I encourage you to maybe write out 00:31:08.970 --> 00:31:12.360 align:middle line:90% phi 1, what that actually is. 00:31:12.360 --> 00:31:17.130 align:middle line:84% I mean, there's-- in fact here, I'll do that. 00:31:17.130 --> 00:31:20.070 align:middle line:90% 00:31:20.070 --> 00:31:22.800 align:middle line:84% I drew the picture that goes over here. 00:31:22.800 --> 00:31:25.110 align:middle line:84% Let me write out what phi 1 actually looks like. 00:31:25.110 --> 00:31:29.550 align:middle line:84% Phi 1 is equal to 0 times the indicator 00:31:29.550 --> 00:31:37.620 align:middle line:84% function where f is between 0 and 1/2 00:31:37.620 --> 00:31:42.060 align:middle line:84% plus a 1/2 times the indicator function where 00:31:42.060 --> 00:31:54.040 align:middle line:84% f is between 1/2 and 1 plus 3/2 times the indicator 00:31:54.040 --> 00:31:58.210 align:middle line:90% function of when f is between-- 00:31:58.210 --> 00:31:59.248 align:middle line:90% no, that's not 3/2. 00:31:59.248 --> 00:32:00.040 align:middle line:90% That should be 1/2. 00:32:00.040 --> 00:32:03.780 align:middle line:90% 00:32:03.780 --> 00:32:05.460 align:middle line:90% No, no, no, what am I doing, 1-- 00:32:05.460 --> 00:32:08.550 align:middle line:90% 00:32:08.550 --> 00:32:13.980 align:middle line:84% times indicator function where f is between 1 and 3/2 00:32:13.980 --> 00:32:20.340 align:middle line:84% plus 3/2 times chi f inverse of where 00:32:20.340 --> 00:32:26.340 align:middle line:90% f is between now 3/2 and 2. 00:32:26.340 --> 00:32:29.040 align:middle line:90% 00:32:29.040 --> 00:32:33.180 align:middle line:84% So 2 is how high up I break up the range. 00:32:33.180 --> 00:32:36.040 align:middle line:84% And then plus this last part, this fn part, 00:32:36.040 --> 00:32:40.260 align:middle line:90% which is 2 times chi, where f-- 00:32:40.260 --> 00:32:47.020 align:middle line:84% the indicator function of where f is bigger than 2 to the n, 00:32:47.020 --> 00:32:48.940 align:middle line:90% So bigger than 2. 00:32:48.940 --> 00:33:01.430 align:middle line:90% 00:33:01.430 --> 00:33:07.130 align:middle line:84% So all of these sets are disjoint. 00:33:07.130 --> 00:33:10.580 align:middle line:84% The Ekn's and fn's for k different from k 00:33:10.580 --> 00:33:14.750 align:middle line:90% prime and fixed n are disjoint. 00:33:14.750 --> 00:33:16.500 align:middle line:84% And they are disjoint from this set. 00:33:16.500 --> 00:33:20.270 align:middle line:84% So this is a simple function, the finitely many values 00:33:20.270 --> 00:33:23.870 align:middle line:84% it takes is 2 to the n along with k times 2 to the minus n. 00:33:23.870 --> 00:33:25.700 align:middle line:84% Or at least the finitely many values 00:33:25.700 --> 00:33:27.840 align:middle line:90% it can take is a subset of that. 00:33:27.840 --> 00:33:30.980 align:middle line:84% So this is a simple function for each n. 00:33:30.980 --> 00:33:34.340 align:middle line:90% 00:33:34.340 --> 00:33:41.080 align:middle line:84% And by design it's always sitting below f. 00:33:41.080 --> 00:33:45.680 align:middle line:84% So let me in fact bring it up here. 00:33:45.680 --> 00:33:48.040 align:middle line:84% Well, let me make the statement and then I'll say. 00:33:48.040 --> 00:34:01.010 align:middle line:84% So by definition for all x in E, En is non-negative 00:34:01.010 --> 00:34:03.860 align:middle line:90% and always sits below f of x. 00:34:03.860 --> 00:34:10.012 align:middle line:90% So how do we see that? 00:34:10.012 --> 00:34:11.679 align:middle line:84% You can see it from the general formula, 00:34:11.679 --> 00:34:13.699 align:middle line:84% but I'll just indicate why just for safety. 00:34:13.699 --> 00:34:17.050 align:middle line:84% One, let's say-- so x has to be in one of these sets. 00:34:17.050 --> 00:34:21.940 align:middle line:84% Let's say it's here where f is between 1/2 and 1. 00:34:21.940 --> 00:34:28.800 align:middle line:84% Then phi-- 1 of x is equal to 1/2 which is less than f 00:34:28.800 --> 00:34:32.280 align:middle line:84% of x because f takes on the value between 1/2 and 1. 00:34:32.280 --> 00:34:41.630 align:middle line:90% 00:34:41.630 --> 00:34:44.704 align:middle line:84% In fact, here, I'll give a the brief argument here. 00:34:44.704 --> 00:34:47.310 align:middle line:90% 00:34:47.310 --> 00:34:52.800 align:middle line:84% If x is in Ekn then by definition 00:34:52.800 --> 00:34:57.780 align:middle line:84% this means that k2 to the minus n is less than f 00:34:57.780 --> 00:35:05.550 align:middle line:84% of x is less than or equal to k plus 1 2 the minus n, which 00:35:05.550 --> 00:35:10.590 align:middle line:84% implies that phi n of x, which is by definition k times 2 00:35:10.590 --> 00:35:11.970 align:middle line:90% to the minus n-- 00:35:11.970 --> 00:35:15.060 align:middle line:84% just, again, by how we've defined the phi sub n's-- where 00:35:15.060 --> 00:35:16.710 align:middle line:90% did we define the phi sub n's-- 00:35:16.710 --> 00:35:22.190 align:middle line:90% there is less than f of x. 00:35:22.190 --> 00:35:44.720 align:middle line:90% 00:35:44.720 --> 00:35:47.090 align:middle line:90% So that's for x and Ekn. 00:35:47.090 --> 00:35:55.160 align:middle line:84% And if x is in f sub n, then that 00:35:55.160 --> 00:36:02.480 align:middle line:84% means f of x is greater than 2 to the n, which is always 00:36:02.480 --> 00:36:05.420 align:middle line:84% bigger than or equal to the phi n of x-- well, 00:36:05.420 --> 00:36:09.710 align:middle line:84% I mean this is actually equal to phi of x. 00:36:09.710 --> 00:36:13.880 align:middle line:84% So we always have-- the phi n's are non-negative. 00:36:13.880 --> 00:36:15.890 align:middle line:84% And they always sit below f of x. 00:36:15.890 --> 00:36:20.090 align:middle line:84% So now let's prove that they are, in fact, increasing, 00:36:20.090 --> 00:36:28.320 align:middle line:90% so part A. 00:36:28.320 --> 00:36:33.735 align:middle line:84% So now we're proving part A, the phi n's increase to f. 00:36:33.735 --> 00:36:39.490 align:middle line:90% 00:36:39.490 --> 00:36:46.450 align:middle line:84% And, again, for fixed n, Ekn and fn, 00:36:46.450 --> 00:36:52.270 align:middle line:84% the Ekn's, and fn form a disjoint union of E. 00:36:52.270 --> 00:36:57.130 align:middle line:84% I just need to check that what I want holds on each 00:36:57.130 --> 00:36:58.570 align:middle line:90% of the Ekn's and fn's. 00:36:58.570 --> 00:37:09.750 align:middle line:84% So suppose x is in Ekn, then f of x 00:37:09.750 --> 00:37:16.140 align:middle line:84% is less than or equal to k plus 1 times 2 to the minus n. 00:37:16.140 --> 00:37:22.100 align:middle line:84% n is bigger than k 2 to the minus n, 00:37:22.100 --> 00:37:27.410 align:middle line:84% and which by a silly trick of just multiplying and dividing 00:37:27.410 --> 00:37:32.870 align:middle line:84% by 2 tells me that f is between 2 times k times 2 to the minus 00:37:32.870 --> 00:37:36.980 align:middle line:84% n minus 1 is less than f of x is less than 00:37:36.980 --> 00:37:42.050 align:middle line:84% or equal to 2k plus 2 times 2 to the minus 00:37:42.050 --> 00:37:49.580 align:middle line:84% n minus 1, which implies that x is 00:37:49.580 --> 00:38:00.350 align:middle line:84% in the union of E2k n plus 1, 2k n plus 1 union 2K 00:38:00.350 --> 00:38:03.590 align:middle line:90% plus 2 n plus 1. 00:38:03.590 --> 00:38:10.130 align:middle line:90% 00:38:10.130 --> 00:38:17.180 align:middle line:84% If x is in E2k n plus 1, then I get that phi n of x 00:38:17.180 --> 00:38:20.930 align:middle line:84% is equal to, by definition, k times 00:38:20.930 --> 00:38:26.360 align:middle line:84% 2 to the minus n, which equals 2k 2 to the minus 00:38:26.360 --> 00:38:32.090 align:middle line:84% n minus 1, which because x is in E 2k n plus 1, 00:38:32.090 --> 00:38:36.395 align:middle line:84% this is equal to phi n plus 1 of x. 00:38:36.395 --> 00:38:41.210 align:middle line:90% 00:38:41.210 --> 00:38:51.300 align:middle line:84% And if x is in E2k plus 2 n plus 1-- 00:38:51.300 --> 00:38:53.750 align:middle line:84% no, I shouldn't-- this shouldn't be a 2. 00:38:53.750 --> 00:38:56.230 align:middle line:90% This should be a 1, I'm sorry-- 00:38:56.230 --> 00:39:01.030 align:middle line:84% because this goes from 2k up to 2k plus 1, and then from 2k 00:39:01.030 --> 00:39:02.860 align:middle line:90% plus 1 up to 2k plus 2. 00:39:02.860 --> 00:39:07.390 align:middle line:84% So it's either in E2k or 2k plus 1. 00:39:07.390 --> 00:39:20.410 align:middle line:84% And if E is in a 2k plus 1, then the n of x is still-- 00:39:20.410 --> 00:39:24.880 align:middle line:84% I mean, x is still in Ekn, so the n 00:39:24.880 --> 00:39:29.170 align:middle line:84% of x is equal to still k times 2 to the minus n, which 00:39:29.170 --> 00:39:33.160 align:middle line:84% is equal to 2k times 2 to the minus n minus 1, which 00:39:33.160 --> 00:39:40.480 align:middle line:84% is less than 2k plus 1 times 2 to the minus n minus 1, 00:39:40.480 --> 00:39:47.320 align:middle line:84% which is by definition, since x is in E2k plus 1, n plus 1 n 00:39:47.320 --> 00:39:48.670 align:middle line:90% plus 1 of x. 00:39:48.670 --> 00:40:01.351 align:middle line:90% 00:40:01.351 --> 00:40:11.430 align:middle line:84% And, similarly, if x is in Fn, then the n of x 00:40:11.430 --> 00:40:17.160 align:middle line:84% is less than or equal to phi n plus 1 of x. 00:40:17.160 --> 00:40:27.960 align:middle line:84% So we verified for all x since E is equal to this union over k 00:40:27.960 --> 00:40:38.250 align:middle line:84% equals 0 2 to the 2n minus 1 Ekn union Fn. 00:40:38.250 --> 00:40:41.450 align:middle line:90% 00:40:41.450 --> 00:40:44.990 align:middle line:84% This implies that for all x in E, 00:40:44.990 --> 00:40:52.445 align:middle line:84% phi n of x is less than or equal to phi n plus 1 of x. 00:40:52.445 --> 00:41:08.930 align:middle line:90% 00:41:08.930 --> 00:41:15.080 align:middle line:90% And this proves A. 00:41:15.080 --> 00:41:20.630 align:middle line:90% Now, how to prove B and C-- 00:41:20.630 --> 00:41:23.360 align:middle line:84% these things will follow from part A in a simple estimate 00:41:23.360 --> 00:41:24.500 align:middle line:90% that we're going to prove. 00:41:24.500 --> 00:41:32.280 align:middle line:90% 00:41:32.280 --> 00:41:37.010 align:middle line:84% So B and C will follow immediately 00:41:37.010 --> 00:41:47.370 align:middle line:84% from the following claim in part a, which the claim is that 00:41:47.370 --> 00:42:00.280 align:middle line:84% for all x in set y in E that f of y is less than or equal to 2 00:42:00.280 --> 00:42:00.850 align:middle line:90% to the n. 00:42:00.850 --> 00:42:04.840 align:middle line:90% 00:42:04.840 --> 00:42:10.090 align:middle line:84% This part we already know that f of x minus phi n of x 00:42:10.090 --> 00:42:11.890 align:middle line:90% is non-negative. 00:42:11.890 --> 00:42:17.330 align:middle line:84% But, in fact, this is bounded by 2 to the minus n. 00:42:17.330 --> 00:42:25.720 align:middle line:90% 00:42:25.720 --> 00:42:45.490 align:middle line:84% So then B and C from A and this claim. 00:42:45.490 --> 00:42:50.280 align:middle line:84% Why does B follow from this claim? 00:42:50.280 --> 00:43:02.740 align:middle line:90% Well, wherever x is-- 00:43:02.740 --> 00:43:10.230 align:middle line:90% 00:43:10.230 --> 00:43:12.750 align:middle line:84% I don't want to have to also explain what happens 00:43:12.750 --> 00:43:14.580 align:middle line:90% if f is equal to infinity. 00:43:14.580 --> 00:43:19.410 align:middle line:84% That also follows basically from the definitions, 00:43:19.410 --> 00:43:20.910 align:middle line:84% not necessarily from this estimate. 00:43:20.910 --> 00:43:24.510 align:middle line:84% But the more important part is when f is finite. 00:43:24.510 --> 00:43:29.340 align:middle line:84% So let's assume f is just finite for every x. 00:43:29.340 --> 00:43:33.180 align:middle line:84% So then every x in E is eventually 00:43:33.180 --> 00:43:35.460 align:middle line:90% in one of these sets. 00:43:35.460 --> 00:43:40.020 align:middle line:90% So there exist let x be fixed. 00:43:40.020 --> 00:43:45.660 align:middle line:84% Then for n sufficiently large, x is in one of these sets 00:43:45.660 --> 00:43:49.140 align:middle line:90% since f of x is finite. 00:43:49.140 --> 00:43:50.430 align:middle line:90% And therefore for all n-- 00:43:50.430 --> 00:43:53.100 align:middle line:90% 00:43:53.100 --> 00:43:58.550 align:middle line:84% for all capital N sufficiently large, 00:43:58.550 --> 00:44:01.670 align:middle line:84% this minus phi n of x is less than or equal to 2 00:44:01.670 --> 00:44:02.750 align:middle line:90% to the minus n. 00:44:02.750 --> 00:44:06.140 align:middle line:84% But then f of x minus phi to the m 00:44:06.140 --> 00:44:10.010 align:middle line:84% is also less than or equal to 2 to the minus n for every m 00:44:10.010 --> 00:44:12.020 align:middle line:84% bigger than or equal to n because the phi 00:44:12.020 --> 00:44:15.080 align:middle line:90% n's are increasing. 00:44:15.080 --> 00:44:17.840 align:middle line:84% If phi n is this close to f, then phi m 00:44:17.840 --> 00:44:21.620 align:middle line:84% is also that close to f if m is bigger than or equal to n, 00:44:21.620 --> 00:44:25.910 align:middle line:84% again, because they're increasing. 00:44:25.910 --> 00:44:30.650 align:middle line:84% So that proves-- that's why pointwise convergence 00:44:30.650 --> 00:44:32.810 align:middle line:90% B follows from this estimate. 00:44:32.810 --> 00:44:35.960 align:middle line:84% As far as part C, that also follows from this estimate 00:44:35.960 --> 00:44:40.700 align:middle line:84% since if I have a fixed B, then I 00:44:40.700 --> 00:44:43.580 align:middle line:84% can choose a natural number just depending on what 00:44:43.580 --> 00:44:46.615 align:middle line:84% B is so that that's set depending 00:44:46.615 --> 00:44:48.740 align:middle line:84% on B and the original statement I'm pointing at it, 00:44:48.740 --> 00:44:50.782 align:middle line:84% but I don't think you can see it from the camera. 00:44:50.782 --> 00:44:55.070 align:middle line:84% That set which depends on B is contained in one of these sets, 00:44:55.070 --> 00:44:59.690 align:middle line:84% and therefore for all x in the set where f is bounded by B, 00:44:59.690 --> 00:45:02.383 align:middle line:90% this holds uniformly in x. 00:45:02.383 --> 00:45:04.550 align:middle line:84% And that's where the uniform convergence comes from. 00:45:04.550 --> 00:45:07.440 align:middle line:90% 00:45:07.440 --> 00:45:11.360 align:middle line:84% But the whole point is that this is the estimate that gives us 00:45:11.360 --> 00:45:16.790 align:middle line:90% B and C once we've proved A. 00:45:16.790 --> 00:45:18.620 align:middle line:90% So let's prove this claim. 00:45:18.620 --> 00:45:20.390 align:middle line:90% It's not hard. 00:45:20.390 --> 00:45:38.690 align:middle line:84% It's basically because we are cutting up the range 00:45:38.690 --> 00:45:44.240 align:middle line:84% not only to height 2 to the n, but with resolution 2 00:45:44.240 --> 00:45:48.875 align:middle line:84% to the minus n at each stage, at each n. 00:45:48.875 --> 00:45:51.960 align:middle line:90% 00:45:51.960 --> 00:46:07.020 align:middle line:84% So to prove the claim we have that the set of y's in E such 00:46:07.020 --> 00:46:11.490 align:middle line:84% that f of x is less than or equal to 2 to the n, 00:46:11.490 --> 00:46:18.510 align:middle line:84% this is equal to the union k equals 0 2 to the 2n minus 1 00:46:18.510 --> 00:46:22.300 align:middle line:90% of the Ekn's. 00:46:22.300 --> 00:46:23.790 align:middle line:84% So if I want to check that bound, 00:46:23.790 --> 00:46:28.920 align:middle line:84% I just have to check it for each, if x is in one of these. 00:46:28.920 --> 00:46:40.140 align:middle line:84% So suppose x is in E to the kn, then-- 00:46:40.140 --> 00:46:42.210 align:middle line:84% I mean, it really is just following from the fact 00:46:42.210 --> 00:46:48.255 align:middle line:84% that we're cutting up the range with resolution 2 00:46:48.255 --> 00:46:49.710 align:middle line:90% to the minus n. 00:46:49.710 --> 00:46:53.250 align:middle line:84% So I'm just going to draw a small little picture here. 00:46:53.250 --> 00:46:57.240 align:middle line:90% We have the axis here. 00:46:57.240 --> 00:47:01.890 align:middle line:84% And here's k plus 1 times 2 to the minus n. 00:47:01.890 --> 00:47:04.500 align:middle line:90% Here's k times 2 to the minus n. 00:47:04.500 --> 00:47:09.790 align:middle line:84% If x is in there, that means there's a-- 00:47:09.790 --> 00:47:12.690 align:middle line:90% 00:47:12.690 --> 00:47:16.290 align:middle line:84% we're looking at the portion of f that sits between k 2 00:47:16.290 --> 00:47:23.500 align:middle line:84% to the minus n and k plus 1 2 to the minus n, then k times 00:47:23.500 --> 00:47:27.910 align:middle line:84% 2 to the minus n is less than f of x is less than or equal to k 00:47:27.910 --> 00:47:33.010 align:middle line:90% plus 1 2 to the minus n. 00:47:33.010 --> 00:47:37.200 align:middle line:84% And remember the simple function on this piece 00:47:37.200 --> 00:47:42.230 align:middle line:90% evaluated in here-- 00:47:42.230 --> 00:47:44.950 align:middle line:90% so here's the x-- 00:47:44.950 --> 00:47:50.730 align:middle line:84% gives the value at the lower bound. 00:47:50.730 --> 00:47:57.080 align:middle line:90% 00:47:57.080 --> 00:48:04.470 align:middle line:84% And therefore we get that fn of x minus phi n of x, this 00:48:04.470 --> 00:48:05.520 align:middle line:90% is equal to-- 00:48:05.520 --> 00:48:10.380 align:middle line:84% not fn, sorry-- f of x minus k2 to the minus 1. 00:48:10.380 --> 00:48:12.780 align:middle line:90% Now x, again, is an Ekn. 00:48:12.780 --> 00:48:15.370 align:middle line:84% So f of x is between these two numbers. 00:48:15.370 --> 00:48:20.580 align:middle line:84% So this is less than or equal to k plus 1 times 2 00:48:20.580 --> 00:48:24.720 align:middle line:84% to the minus n minus k 2 to the minus n. 00:48:24.720 --> 00:48:26.550 align:middle line:84% And this equals 2 to the minus n. 00:48:26.550 --> 00:48:31.120 align:middle line:90% 00:48:31.120 --> 00:48:35.380 align:middle line:84% So whenever we have an x in one of these sets, 00:48:35.380 --> 00:48:39.550 align:middle line:84% it has to be within 2 to the minus n. 00:48:39.550 --> 00:48:41.710 align:middle line:84% The simple function evaluated at that x 00:48:41.710 --> 00:48:44.710 align:middle line:84% has to be within 2 to the minus n of f. 00:48:44.710 --> 00:48:51.020 align:middle line:84% This is just by construction, by how we've cut up the y-axis, 00:48:51.020 --> 00:48:53.110 align:middle line:90% if you like, the range. 00:48:53.110 --> 00:48:58.610 align:middle line:84% We're cutting it up with resolution 2 to the minus n. 00:48:58.610 --> 00:49:02.420 align:middle line:84% All right, so that proves the claim which, as I said along 00:49:02.420 --> 00:49:12.050 align:middle line:90% with part A, proves B and C. 00:49:12.050 --> 00:49:14.480 align:middle line:84% So that proves that every measurable function 00:49:14.480 --> 00:49:17.480 align:middle line:84% is almost at least non-negative extended 00:49:17.480 --> 00:49:20.060 align:middle line:84% real valued measurable function is 00:49:20.060 --> 00:49:26.610 align:middle line:84% a limit of a sequence of simple functions. 00:49:26.610 --> 00:49:31.220 align:middle line:84% Now, this theorem carries over without difficulty 00:49:31.220 --> 00:49:33.260 align:middle line:84% to complex value functions after I just 00:49:33.260 --> 00:49:37.710 align:middle line:84% introduce a breakup of a function in general. 00:49:37.710 --> 00:49:49.680 align:middle line:84% So if E is a function from minus infinity to infinity, 00:49:49.680 --> 00:49:55.870 align:middle line:84% so an extended real-value function now, 00:49:55.870 --> 00:50:02.350 align:middle line:84% we define its positive and negative part f 00:50:02.350 --> 00:50:09.630 align:middle line:84% plus of x to equal the max of f of x, 0. 00:50:09.630 --> 00:50:11.940 align:middle line:84% So this is the positive part of F f. 00:50:11.940 --> 00:50:20.410 align:middle line:90% 00:50:20.410 --> 00:50:27.062 align:middle line:84% f minus of x is equal to the min of-- 00:50:27.062 --> 00:50:27.895 align:middle line:90% the max, I'm sorry-- 00:50:27.895 --> 00:50:31.350 align:middle line:90% 00:50:31.350 --> 00:50:34.800 align:middle line:90% minus f of x, 0. 00:50:34.800 --> 00:50:39.720 align:middle line:90% This is the negative part of f. 00:50:39.720 --> 00:50:48.040 align:middle line:90% 00:50:48.040 --> 00:50:51.100 align:middle line:84% So what about these positive and negative parts? 00:50:51.100 --> 00:50:57.960 align:middle line:84% Then f is equal to f plus minus F minus. 00:50:57.960 --> 00:50:59.010 align:middle line:90% You can just check that. 00:50:59.010 --> 00:51:02.040 align:middle line:84% I mean, take any x if f is positive or non-negative 00:51:02.040 --> 00:51:05.460 align:middle line:90% on that, then I get f of x out. 00:51:05.460 --> 00:51:10.180 align:middle line:84% If it is negative, then I will get 00:51:10.180 --> 00:51:13.480 align:middle line:84% minus f of x, which is the absolute value times minus 00:51:13.480 --> 00:51:16.840 align:middle line:90% gives me back f of x. 00:51:16.840 --> 00:51:24.760 align:middle line:84% And the absolute value of f is equal to f plus plus f minus. 00:51:24.760 --> 00:51:34.520 align:middle line:90% 00:51:34.520 --> 00:51:37.310 align:middle line:84% So let me make a comment that if-- 00:51:37.310 --> 00:51:40.970 align:middle line:84% this is just a definition for an arbitrary function from E 00:51:40.970 --> 00:51:42.710 align:middle line:90% to the extended real numbers-- 00:51:42.710 --> 00:51:45.800 align:middle line:84% if f is measurable, then each of these functions 00:51:45.800 --> 00:51:47.720 align:middle line:84% is measurable because this is-- if you 00:51:47.720 --> 00:51:50.780 align:middle line:84% like the supremum of the sequence of functions 00:51:50.780 --> 00:51:53.580 align:middle line:84% given by f and then 0 afterwards. 00:51:53.580 --> 00:51:56.660 align:middle line:84% And this is given by the supremum of functions 00:51:56.660 --> 00:51:59.480 align:middle line:84% given by minus f and 0 afterwards. 00:51:59.480 --> 00:52:03.440 align:middle line:84% So if f is measurable, it's positive. 00:52:03.440 --> 00:52:06.290 align:middle line:84% And negative parts are also measurable functions. 00:52:06.290 --> 00:52:08.630 align:middle line:90% And they're also non-negative. 00:52:08.630 --> 00:52:16.410 align:middle line:84% So maybe that wasn't clear, or at least it should be clear. 00:52:16.410 --> 00:52:19.850 align:middle line:84% This is the max always involving 0, so it's always non-negative. 00:52:19.850 --> 00:52:29.130 align:middle line:90% 00:52:29.130 --> 00:52:32.610 align:middle line:84% So now the construction we did a minute ago 00:52:32.610 --> 00:52:39.270 align:middle line:84% essentially carries over to the case of complex value 00:52:39.270 --> 00:52:40.660 align:middle line:90% measurable functions. 00:52:40.660 --> 00:52:53.280 align:middle line:84% So let E be measurable and f from E to C be measurable. 00:52:53.280 --> 00:52:57.760 align:middle line:90% 00:52:57.760 --> 00:53:09.570 align:middle line:84% Then there exists sequence of simple functions 00:53:09.570 --> 00:53:15.230 align:middle line:90% phi n such that-- 00:53:15.230 --> 00:53:16.850 align:middle line:84% analogs of those three properties 00:53:16.850 --> 00:53:28.610 align:middle line:84% hold, A is for all x in E. These are increasing in modulus, 00:53:28.610 --> 00:53:29.480 align:middle line:90% in absolute value. 00:53:29.480 --> 00:53:31.580 align:middle line:90% 0 is bigger than or equal to-- 00:53:31.580 --> 00:53:37.750 align:middle line:84% of course ph 0 of x is less than or equal to the absolute value 00:53:37.750 --> 00:53:40.330 align:middle line:84% of phi 1 of x less than or equal to the, 00:53:40.330 --> 00:53:42.805 align:middle line:84% let's say absolute value and modulus, the same thing-- 00:53:42.805 --> 00:53:45.490 align:middle line:90% 00:53:45.490 --> 00:53:51.130 align:middle line:84% is less than or equal to the absolute value of f of x. 00:53:51.130 --> 00:54:04.000 align:middle line:90% 00:54:04.000 --> 00:54:08.485 align:middle line:84% These phi n's are converging to f pointwise. 00:54:08.485 --> 00:54:21.870 align:middle line:90% 00:54:21.870 --> 00:54:25.470 align:middle line:84% And, finally, convergence is uniform on sets 00:54:25.470 --> 00:54:26.740 align:middle line:90% where f is bounded. 00:54:26.740 --> 00:54:31.365 align:middle line:84% So for all B bigger than or equal to 0-- 00:54:31.365 --> 00:54:35.952 align:middle line:90% 00:54:35.952 --> 00:54:39.970 align:middle line:90% What did I say a minute ago. 00:54:39.970 --> 00:54:54.660 align:middle line:84% Oh, phi n convergence to f uniformly on the set x in E 00:54:54.660 --> 00:54:59.350 align:middle line:84% such that the absolute value of f 00:54:59.350 --> 00:55:08.440 align:middle line:84% of x, or the modulus of f of x, is less than or equal to B. 00:55:08.440 --> 00:55:14.800 align:middle line:84% So this theorem follows immediately 00:55:14.800 --> 00:55:17.020 align:middle line:84% from the previous theorem because now what I do 00:55:17.020 --> 00:55:18.040 align:middle line:90% is I just take f. 00:55:18.040 --> 00:55:20.650 align:middle line:84% I split it into its real and imaginary parts. 00:55:20.650 --> 00:55:23.770 align:middle line:84% And then I split the imaginary-- the real and imaginary parts 00:55:23.770 --> 00:55:25.630 align:middle line:84% into both positive and negative parts. 00:55:25.630 --> 00:55:30.640 align:middle line:84% So I will leave it to you to actually fill in the details, 00:55:30.640 --> 00:55:40.800 align:middle line:84% but you apply the previous theorem 00:55:40.800 --> 00:55:46.020 align:middle line:84% to the real part of f plus or minus, 00:55:46.020 --> 00:55:50.460 align:middle line:84% which are now non-negative measurable functions. 00:55:50.460 --> 00:55:54.740 align:middle line:90% 00:55:54.740 --> 00:55:56.570 align:middle line:84% And the positive and negative parts 00:55:56.570 --> 00:56:02.290 align:middle line:84% of the imaginary part of f, again, which are non-negative 00:56:02.290 --> 00:56:04.120 align:middle line:84% measurable functions, and then you 00:56:04.120 --> 00:56:11.430 align:middle line:84% just take linear combinations of these simple functions that 00:56:11.430 --> 00:56:12.465 align:middle line:90% add up to f. 00:56:12.465 --> 00:56:15.590 align:middle line:90% 00:56:15.590 --> 00:56:18.090 align:middle line:84% You will take the sequence of simple functions corresponding 00:56:18.090 --> 00:56:21.900 align:middle line:84% to the real part of f and subtract the sequence 00:56:21.900 --> 00:56:24.570 align:middle line:84% of simple functions that you got for the minus 00:56:24.570 --> 00:56:27.960 align:middle line:84% for the negative part of f, real part of f. 00:56:27.960 --> 00:56:34.263 align:middle line:84% And you will take that and add i times the positive part 00:56:34.263 --> 00:56:36.180 align:middle line:84% or the sequence of simple functions converging 00:56:36.180 --> 00:56:39.360 align:middle line:84% to the positive part of the imaginary part of f 00:56:39.360 --> 00:56:42.600 align:middle line:84% minus the sequence of functions, simple functions converging 00:56:42.600 --> 00:56:46.170 align:middle line:84% to the negative part of the imaginary part of f. 00:56:46.170 --> 00:56:48.780 align:middle line:84% So that's what I mean by apply previous theorem 00:56:48.780 --> 00:56:51.780 align:middle line:84% to the positive and negative parts 00:56:51.780 --> 00:56:56.150 align:middle line:84% of the real and imaginary parts of f. 00:56:56.150 --> 00:56:59.450 align:middle line:84% So what's the significance of this theorem? 00:56:59.450 --> 00:57:04.830 align:middle line:90% Not only-- let me just say if-- 00:57:04.830 --> 00:57:07.650 align:middle line:84% also showing that somehow measurable functions 00:57:07.650 --> 00:57:09.870 align:middle line:84% are well approximated or almost simple functions, 00:57:09.870 --> 00:57:13.920 align:middle line:84% this also gives us a way of maybe defining the integral 00:57:13.920 --> 00:57:16.980 align:middle line:84% at least of non-negative functions that way 00:57:16.980 --> 00:57:21.630 align:middle line:84% we don't have to deal with possible deals 00:57:21.630 --> 00:57:26.550 align:middle line:84% with subtracting infinity from infinity, 00:57:26.550 --> 00:57:31.200 align:middle line:84% but by simply defining its integral 00:57:31.200 --> 00:57:38.330 align:middle line:84% to be the limit of these integrals 00:57:38.330 --> 00:57:39.860 align:middle line:90% of these simple functions. 00:57:39.860 --> 00:57:43.280 align:middle line:84% And for a simple function, we would presumably 00:57:43.280 --> 00:57:44.810 align:middle line:90% know how to define an integral. 00:57:44.810 --> 00:57:47.810 align:middle line:84% It would just be the numbers time the measure of the sets 00:57:47.810 --> 00:57:51.080 align:middle line:84% that appear in the indicator functions 00:57:51.080 --> 00:57:52.610 align:middle line:90% for these simple functions. 00:57:52.610 --> 00:57:56.010 align:middle line:90% 00:57:56.010 --> 00:57:58.080 align:middle line:84% And like I said last lecture, if you 00:57:58.080 --> 00:58:00.930 align:middle line:84% wanted to define the Lebesgue integral that way, 00:58:00.930 --> 00:58:03.630 align:middle line:84% you would run up against, well, does this number 00:58:03.630 --> 00:58:06.600 align:middle line:84% depend on the sequence of simple functions 00:58:06.600 --> 00:58:09.530 align:middle line:90% you chose to approximate f? 00:58:09.530 --> 00:58:11.280 align:middle line:84% But we're not going to define the Lebesgue 00:58:11.280 --> 00:58:12.155 align:middle line:90% integral in that way. 00:58:12.155 --> 00:58:14.910 align:middle line:84% We're going to define it a little bit differently, which 00:58:14.910 --> 00:58:16.750 align:middle line:84% is what we're going to move on to now, 00:58:16.750 --> 00:58:20.655 align:middle line:84% which is the Lebesgue integral of a non-negative function. 00:58:20.655 --> 00:58:23.220 align:middle line:90% 00:58:23.220 --> 00:58:29.190 align:middle line:84% And then we will define the Lebesgue integral, or Lebesgue 00:58:29.190 --> 00:58:30.300 align:middle line:90% integrable functions. 00:58:30.300 --> 00:58:33.030 align:middle line:84% These will be complex value functions now for which 00:58:33.030 --> 00:58:35.700 align:middle line:90% we can define an integral for. 00:58:35.700 --> 00:58:38.130 align:middle line:90% And that is the full theory of-- 00:58:38.130 --> 00:58:45.880 align:middle line:84% and that's the end of the game for as far as defining 00:58:45.880 --> 00:58:46.670 align:middle line:90% Lebesgue integral. 00:58:46.670 --> 00:58:50.530 align:middle line:84% And then we'll prove some convergence theorems 00:58:50.530 --> 00:58:53.770 align:middle line:84% along the way, which make Lebesgue integral stronger 00:58:53.770 --> 00:58:55.080 align:middle line:90% than Riemann integral. 00:58:55.080 --> 00:59:04.360 align:middle line:84% So now we're moving on to the Lebesgue integral 00:59:04.360 --> 00:59:05.860 align:middle line:90% of a non-negative function. 00:59:05.860 --> 00:59:15.120 align:middle line:90% 00:59:15.120 --> 00:59:18.360 align:middle line:84% And why start with a non-negative function? 00:59:18.360 --> 00:59:21.475 align:middle line:84% Again, because I just pulled this trick on you a minute ago, 00:59:21.475 --> 00:59:23.100 align:middle line:84% that if somehow we know how to do stuff 00:59:23.100 --> 00:59:26.460 align:middle line:84% for non-negative measurable functions, 00:59:26.460 --> 00:59:28.860 align:middle line:84% then by playing this game where we 00:59:28.860 --> 00:59:30.420 align:middle line:84% take the real and imaginary parts 00:59:30.420 --> 00:59:33.090 align:middle line:84% and split it up into positive and negative parts, 00:59:33.090 --> 00:59:35.740 align:middle line:84% hopefully we can do something for general functions. 00:59:35.740 --> 00:59:41.098 align:middle line:84% So that's why we start out with introducing or defining 00:59:41.098 --> 00:59:43.140 align:middle line:84% the Lebesgue integral of a non-negative function. 00:59:43.140 --> 00:59:45.700 align:middle line:90% 00:59:45.700 --> 00:59:56.940 align:middle line:84% So definition, if E is a measurable subset of R, 00:59:56.940 --> 01:00:05.910 align:middle line:84% we define L plus of E. This is a set of all extended real-value 01:00:05.910 --> 01:00:09.240 align:middle line:84% functions, non-negative real-value functions that are 01:00:09.240 --> 01:00:09.960 align:middle line:90% measurable . 01:00:09.960 --> 01:00:20.040 align:middle line:90% 01:00:20.040 --> 01:00:23.820 align:middle line:84% And now what the goal is is to define 01:00:23.820 --> 01:00:28.680 align:middle line:84% the integral of a function like that's in this class. 01:00:28.680 --> 01:00:30.780 align:middle line:84% And it may be an infinite number. 01:00:30.780 --> 01:00:32.590 align:middle line:90% It may not be. 01:00:32.590 --> 01:00:34.980 align:middle line:84% And in order to do that we're first 01:00:34.980 --> 01:00:38.190 align:middle line:84% going to define how to integrate the simplest 01:00:38.190 --> 01:00:39.640 align:middle line:90% type of functions-- 01:00:39.640 --> 01:00:40.860 align:middle line:90% well, simple functions. 01:00:40.860 --> 01:00:45.480 align:middle line:90% 01:00:45.480 --> 01:00:51.395 align:middle line:90% So let phi be a simple function. 01:00:51.395 --> 01:00:55.850 align:middle line:90% 01:00:55.850 --> 01:00:58.700 align:middle line:84% And let's write phi in this canonical way. 01:00:58.700 --> 01:01:01.830 align:middle line:90% 01:01:01.830 --> 01:01:15.610 align:middle line:84% So phi is equal to a sum of aj chi Aj 01:01:15.610 --> 01:01:26.480 align:middle line:90% where for all i not equal to j-- 01:01:26.480 --> 01:01:38.350 align:middle line:84% well, first off, for all j, Aj is a subset of E. 01:01:38.350 --> 01:01:45.143 align:middle line:84% For all i not equal to j, Ai intersect Aj is empty. 01:01:45.143 --> 01:01:46.060 align:middle line:90% So these are disjoint. 01:01:46.060 --> 01:01:49.130 align:middle line:90% 01:01:49.130 --> 01:02:14.346 align:middle line:84% And their union gives me the set E. 01:02:14.346 --> 01:02:25.810 align:middle line:84% The Lebesgue integral of the simple function phi 01:02:25.810 --> 01:02:29.980 align:middle line:84% is the number, which I said is the simplest way, 01:02:29.980 --> 01:02:33.350 align:middle line:84% or what you would expect to define-- 01:02:33.350 --> 01:02:45.540 align:middle line:90% 01:02:45.540 --> 01:02:49.350 align:middle line:84% how to define the integral of a simple function. 01:02:49.350 --> 01:02:55.700 align:middle line:90% So we know how to measure sets. 01:02:55.700 --> 01:03:00.440 align:middle line:84% And remember we initially built up 01:03:00.440 --> 01:03:06.620 align:middle line:84% measure so that the integral, which 01:03:06.620 --> 01:03:09.570 align:middle line:84% would be a theory of area underneath the curve 01:03:09.570 --> 01:03:10.070 align:middle line:90% should be-- 01:03:10.070 --> 01:03:13.430 align:middle line:90% 01:03:13.430 --> 01:03:15.620 align:middle line:84% the integral of the indicator function 01:03:15.620 --> 01:03:19.820 align:middle line:84% should be the area underneath the curve of 1, 01:03:19.820 --> 01:03:22.910 align:middle line:84% area being the measure of the set where that indicator 01:03:22.910 --> 01:03:26.070 align:middle line:90% function is 1. 01:03:26.070 --> 01:03:32.370 align:middle line:84% And therefore by linearity, this will 01:03:32.370 --> 01:03:35.320 align:middle line:84% be how we define the Lebesgue integral of a simple function. 01:03:35.320 --> 01:03:43.470 align:middle line:84% So the loop Lebesgue of phi is the following number, E phi-- 01:03:43.470 --> 01:03:45.240 align:middle line:90% this is how it's defined-- 01:03:45.240 --> 01:03:53.505 align:middle line:84% sum from j equals 1 to n Aj measure of Aj. 01:03:53.505 --> 01:04:03.770 align:middle line:90% 01:04:03.770 --> 01:04:08.990 align:middle line:84% And this number could be infinite. 01:04:08.990 --> 01:04:13.030 align:middle line:90% 01:04:13.030 --> 01:04:19.090 align:middle line:84% All right, and instead of writing just the integral 01:04:19.090 --> 01:04:21.670 align:middle line:84% over E of phi, I might add a dx in there. 01:04:21.670 --> 01:04:26.116 align:middle line:90% 01:04:26.116 --> 01:04:27.855 align:middle line:84% I'm just warning you ahead of time. 01:04:27.855 --> 01:04:36.984 align:middle line:90% 01:04:36.984 --> 01:04:38.980 align:middle line:84% All right, so this is the Lebesgue integral 01:04:38.980 --> 01:04:39.970 align:middle line:90% of a simple function. 01:04:39.970 --> 01:04:42.250 align:middle line:84% Again, we split it up into this canonical way 01:04:42.250 --> 01:04:45.880 align:middle line:84% where it's just the indicator function of disjoint sets 01:04:45.880 --> 01:04:50.360 align:middle line:84% which whose union gives the set of the domain 01:04:50.360 --> 01:04:55.270 align:middle line:84% and where these numbers out in front, these coefficients, 01:04:55.270 --> 01:05:04.420 align:middle line:90% give you the numbers that go-- 01:05:04.420 --> 01:05:05.810 align:middle line:90% of the elements of the range. 01:05:05.810 --> 01:05:10.510 align:middle line:84% So, for example again, I mean, this is-- 01:05:10.510 --> 01:05:16.660 align:middle line:84% let's say A is from-- or let's say E is the interval a, b. 01:05:16.660 --> 01:05:18.880 align:middle line:90% You can check that-- 01:05:18.880 --> 01:05:22.070 align:middle line:90% 01:05:22.070 --> 01:05:24.790 align:middle line:84% well, I mean, it's just from the definition 01:05:24.790 --> 01:05:31.640 align:middle line:84% that if my simple function is, in fact, these sets are just 01:05:31.640 --> 01:05:43.650 align:middle line:84% intervals, so if this is what my simple function looks like, 01:05:43.650 --> 01:05:47.100 align:middle line:84% it takes finitely many values, this one, this one, this one, 01:05:47.100 --> 01:05:49.080 align:middle line:90% this one, this one. 01:05:49.080 --> 01:05:51.990 align:middle line:84% And on these sets whose disjoint union 01:05:51.990 --> 01:06:02.030 align:middle line:84% forms a, b, then as I've defined and since-- 01:06:02.030 --> 01:06:04.400 align:middle line:84% as I've defined the integral and since the measure 01:06:04.400 --> 01:06:08.450 align:middle line:84% of an interval equals the length of the interval, 01:06:08.450 --> 01:06:13.115 align:middle line:84% again, you should count this as some more comments. 01:06:13.115 --> 01:06:23.680 align:middle line:90% 01:06:23.680 --> 01:06:27.790 align:middle line:84% This integral equals the sum of-- 01:06:27.790 --> 01:06:31.220 align:middle line:90% so this would be aj. 01:06:31.220 --> 01:06:36.420 align:middle line:90% 01:06:36.420 --> 01:06:39.540 align:middle line:84% Then how I've defined the Lebesgue integral, 01:06:39.540 --> 01:06:49.720 align:middle line:84% it spits out the area underneath this simple function 01:06:49.720 --> 01:06:55.060 align:middle line:84% that is taking these values on these intervals. 01:06:55.060 --> 01:06:55.900 align:middle line:90% I hope that's clear. 01:06:55.900 --> 01:06:58.233 align:middle line:84% It's towards the end of the day so maybe my explanations 01:06:58.233 --> 01:07:03.580 align:middle line:84% are getting a bit wonky, but I hope it's clear. 01:07:03.580 --> 01:07:13.490 align:middle line:90% 01:07:13.490 --> 01:07:16.130 align:middle line:84% And then we're going to use this definition of simple-- of how 01:07:16.130 --> 01:07:17.380 align:middle line:90% to integrate simple functions. 01:07:17.380 --> 01:07:20.960 align:middle line:84% We're going to extend it to general elements of L 01:07:20.960 --> 01:07:23.600 align:middle line:84% plus of E, the non-negative measurable functions. 01:07:23.600 --> 01:07:29.400 align:middle line:84% But first let's prove a few properties 01:07:29.400 --> 01:07:32.550 align:middle line:84% of how we've defined the integral for simple functions. 01:07:32.550 --> 01:07:45.010 align:middle line:90% 01:07:45.010 --> 01:07:49.120 align:middle line:90% Let's take two simple functions. 01:07:49.120 --> 01:07:54.270 align:middle line:84% First is that if c is bigger than or equal to 0, 01:07:54.270 --> 01:07:59.850 align:middle line:84% then the integral of c times phi over E, 01:07:59.850 --> 01:08:03.960 align:middle line:84% this is equal to c times the integral of E phi. 01:08:03.960 --> 01:08:08.530 align:middle line:90% 01:08:08.530 --> 01:08:20.540 align:middle line:84% Two, the Lebesgue integral over E of phi plus psi 01:08:20.540 --> 01:08:24.020 align:middle line:84% is equal to the Lebesgue integral 01:08:24.020 --> 01:08:30.660 align:middle line:84% of phi plus the Lebesgue integral of psi. 01:08:30.660 --> 01:08:41.439 align:middle line:84% And third property is if phi is less than or equal to psi on E, 01:08:41.439 --> 01:08:43.130 align:middle line:84% so I'm writing that shorthand by that. 01:08:43.130 --> 01:08:45.760 align:middle line:84% So what I'm saying in that statement is for all x in E, 01:08:45.760 --> 01:08:49.450 align:middle line:84% phi of x is less than or equal to psi of x, then 01:08:49.450 --> 01:08:52.960 align:middle line:84% what you expect is the integral of phi 01:08:52.960 --> 01:08:55.344 align:middle line:84% is less than or equal to the integral of psi. 01:08:55.344 --> 01:09:07.609 align:middle line:90% 01:09:07.609 --> 01:09:12.725 align:middle line:84% And let me include one more very simple property. 01:09:12.725 --> 01:09:16.819 align:middle line:90% 01:09:16.819 --> 01:09:24.160 align:middle line:84% If f is-- let's add a little punctuation. 01:09:24.160 --> 01:09:31.899 align:middle line:84% If f is a measurable subset of E, 01:09:31.899 --> 01:09:35.430 align:middle line:84% then phi which is a simple function on E 01:09:35.430 --> 01:09:37.200 align:middle line:90% is also a simple function on f. 01:09:37.200 --> 01:09:40.468 align:middle line:84% It takes only finitely many values on f as well, 01:09:40.468 --> 01:09:42.135 align:middle line:84% and therefore it has an integral over f. 01:09:42.135 --> 01:09:44.990 align:middle line:90% 01:09:44.990 --> 01:09:55.500 align:middle line:84% And my claim is that is equal to the integral over E 01:09:55.500 --> 01:09:58.440 align:middle line:84% of phi times the indicator function of f. 01:09:58.440 --> 01:10:02.850 align:middle line:90% 01:10:02.850 --> 01:10:24.790 align:middle line:84% And that's less than or equal to the integral over E. 01:10:24.790 --> 01:10:27.662 align:middle line:84% Now I will leave problem 4 to you. 01:10:27.662 --> 01:10:29.120 align:middle line:84% I might even put on the assignment. 01:10:29.120 --> 01:10:34.730 align:middle line:84% But it will follow once you've seen how we prove 1, 2, and 3. 01:10:34.730 --> 01:10:37.170 align:middle line:84% You'll be like, OK, so I know how to do this. 01:10:37.170 --> 01:10:45.640 align:middle line:84% So number 1 is pretty easy simply 01:10:45.640 --> 01:10:49.245 align:middle line:84% because multiplying by a non-negative constant 01:10:49.245 --> 01:10:51.370 align:middle line:84% just carries through and just changes the constant, 01:10:51.370 --> 01:10:53.680 align:middle line:90% but not the sets. 01:10:53.680 --> 01:11:02.035 align:middle line:84% So c times phi is equal to c times a sub j. 01:11:02.035 --> 01:11:07.120 align:middle line:90% 01:11:07.120 --> 01:11:12.340 align:middle line:84% And therefore the integral of c times phi over E, 01:11:12.340 --> 01:11:14.980 align:middle line:84% this is by definition equal to j-- 01:11:14.980 --> 01:11:19.440 align:middle line:84% sum from j equals 1 to n of c times a sub j 01:11:19.440 --> 01:11:22.830 align:middle line:90% measure of a sub j. 01:11:22.830 --> 01:11:28.770 align:middle line:84% And this is equal to c times sum from j equals 1 to n 01:11:28.770 --> 01:11:32.220 align:middle line:90% of a sub j m of a sub j. 01:11:32.220 --> 01:11:46.670 align:middle line:84% And this equals c times the integral of phi over E. 01:11:46.670 --> 01:11:51.570 align:middle line:84% All right, so to prove 2 we write 01:11:51.570 --> 01:12:00.390 align:middle line:84% phi in this canonical form, sum over a sub j 01:12:00.390 --> 01:12:04.020 align:middle line:84% times the indicator function of a sub j's 01:12:04.020 --> 01:12:05.820 align:middle line:84% where, again, these sets are disjoint. 01:12:05.820 --> 01:12:08.850 align:middle line:84% And their union gives me E. And then also I 01:12:08.850 --> 01:12:14.430 align:middle line:84% do the same thing for phis, now k equals 1 to m, 01:12:14.430 --> 01:12:19.400 align:middle line:84% maybe take on different number of values, 01:12:19.400 --> 01:12:25.680 align:middle line:84% bk, chi, bk, where again the bk's is are disjoint 01:12:25.680 --> 01:12:50.440 align:middle line:90% and their union gives me E. 01:12:50.440 --> 01:12:56.080 align:middle line:84% So since the union of the Aj's gives me 01:12:56.080 --> 01:13:09.570 align:middle line:84% E and the union of the bk's give me, 01:13:09.570 --> 01:13:16.860 align:middle line:84% this implies that if I want to look at one of these Aj's it's 01:13:16.860 --> 01:13:26.360 align:middle line:84% equal to a union of a certain type, is equal to the union k 01:13:26.360 --> 01:13:33.470 align:middle line:84% equals 1 to m of Aj intersect Bk because this is just 01:13:33.470 --> 01:13:36.230 align:middle line:84% going to be equal to Aj intersect 01:13:36.230 --> 01:13:41.850 align:middle line:84% the union of the Bk's, but the union of the Bk's gives me E, 01:13:41.850 --> 01:13:54.830 align:middle line:84% and similarly for the Bk's because the union of Aj's gives 01:13:54.830 --> 01:14:00.220 align:middle line:90% me E. 01:14:00.220 --> 01:14:03.985 align:middle line:84% And, again, and these unions are disjoint. 01:14:03.985 --> 01:14:13.620 align:middle line:90% 01:14:13.620 --> 01:14:16.530 align:middle line:84% The Aj's are disjoint from each other. 01:14:16.530 --> 01:14:18.490 align:middle line:84% The Bj's are disjoint from each other. 01:14:18.490 --> 01:14:22.110 align:middle line:84% So for all j and k, Aj intersect Bk 01:14:22.110 --> 01:14:27.900 align:middle line:84% is going to be disjoint from another Aj prime intersect Bk 01:14:27.900 --> 01:14:31.170 align:middle line:84% prime when j or k do not equal each other. 01:14:31.170 --> 01:14:38.770 align:middle line:90% 01:14:38.770 --> 01:14:42.880 align:middle line:84% So since these are disjoint unions, 01:14:42.880 --> 01:15:02.090 align:middle line:84% we have from the additivity property of Lebesgue measure, 01:15:02.090 --> 01:15:06.370 align:middle line:84% we get that the integral over phi E 01:15:06.370 --> 01:15:10.630 align:middle line:84% plus the integral over E of psi, this 01:15:10.630 --> 01:15:15.520 align:middle line:84% is equal to, by definition, j equals 1 to n 01:15:15.520 --> 01:15:20.185 align:middle line:90% of Aj measure of Aj plus-- 01:15:20.185 --> 01:15:23.990 align:middle line:90% 01:15:23.990 --> 01:15:30.080 align:middle line:90% equals 1 to m Bk measure of Bk. 01:15:30.080 --> 01:15:34.250 align:middle line:84% Now, Aj is written as this disjoint union. 01:15:34.250 --> 01:15:35.930 align:middle line:84% And therefore the measure of Aj is 01:15:35.930 --> 01:15:48.650 align:middle line:84% equal to the sum of the measure of these, of Aj intersect Bk 01:15:48.650 --> 01:15:49.640 align:middle line:90% plus-- 01:15:49.640 --> 01:15:51.020 align:middle line:90% and then the same thing here. 01:15:51.020 --> 01:15:54.260 align:middle line:84% The measure of-- so Bk is equal to this disjoint union. 01:15:54.260 --> 01:15:57.860 align:middle line:84% So its measure is the sum of the measure. 01:15:57.860 --> 01:16:01.805 align:middle line:90% So this sum now includes j. 01:16:01.805 --> 01:16:09.960 align:middle line:90% 01:16:09.960 --> 01:16:18.790 align:middle line:84% So let me just rewrite this as sum over jk, aj plus bk measure 01:16:18.790 --> 01:16:21.170 align:middle line:90% of age aj bk. 01:16:21.170 --> 01:16:26.920 align:middle line:90% 01:16:26.920 --> 01:16:36.800 align:middle line:84% But the point is that the sum of these two simple functions 01:16:36.800 --> 01:16:41.690 align:middle line:84% you can check, this is equal to the sum over jk 01:16:41.690 --> 01:16:46.220 align:middle line:84% of Aj plus Bk times the indicator 01:16:46.220 --> 01:16:50.045 align:middle line:90% function of Aj intersect Bk. 01:16:50.045 --> 01:16:53.620 align:middle line:90% 01:16:53.620 --> 01:16:58.030 align:middle line:84% So at those x's when phi equals Aj, 01:16:58.030 --> 01:17:03.670 align:middle line:84% and at those x's where phi equals Bk, then I'm in this set 01:17:03.670 --> 01:17:05.140 align:middle line:90% and the two sides agree. 01:17:05.140 --> 01:17:10.350 align:middle line:90% 01:17:10.350 --> 01:17:18.050 align:middle line:84% This implies that the integral over E of phi plus psi 01:17:18.050 --> 01:17:30.330 align:middle line:84% is equal to measure of Aj intersect 01:17:30.330 --> 01:17:38.640 align:middle line:84% Bk, which as we just saw is equal to the sum 01:17:38.640 --> 01:17:39.570 align:middle line:90% of the integrals. 01:17:39.570 --> 01:17:44.440 align:middle line:90% 01:17:44.440 --> 01:17:45.610 align:middle line:90% So that was 2. 01:17:45.610 --> 01:17:50.270 align:middle line:90% 3 is not too difficult either. 01:17:50.270 --> 01:17:53.290 align:middle line:84% So, again, let's assume that phi and psi 01:17:53.290 --> 01:17:58.210 align:middle line:84% are written in that way, in this canonical way. 01:17:58.210 --> 01:18:08.350 align:middle line:84% Then for all x in E, phi of x is less than or equal to psi x. 01:18:08.350 --> 01:18:20.960 align:middle line:84% This is equivalent to aj is less than or equal to bk whenever 01:18:20.960 --> 01:18:25.040 align:middle line:90% aj in bk is nonempty. 01:18:25.040 --> 01:18:33.760 align:middle line:90% 01:18:33.760 --> 01:18:35.950 align:middle line:84% Thus, again, we're going to use the additivity 01:18:35.950 --> 01:18:38.110 align:middle line:84% of the Lebesgue measure and the fact 01:18:38.110 --> 01:18:40.660 align:middle line:84% that the unions of these Bk's give me E, 01:18:40.660 --> 01:18:44.980 align:middle line:84% then if I look at the interval of E of phi, 01:18:44.980 --> 01:18:52.330 align:middle line:84% this is sum j equals 1 to n aj measure of Aj. 01:18:52.330 --> 01:19:02.620 align:middle line:84% This is equal to the sum jk aj times the measure of Aj Bk 01:19:02.620 --> 01:19:06.220 align:middle line:84% because the Aj is a union over Aj intersect Bks. 01:19:06.220 --> 01:19:08.140 align:middle line:90% And that's a disjoint union. 01:19:08.140 --> 01:19:13.880 align:middle line:84% And now whenever this is non-zero, 01:19:13.880 --> 01:19:21.620 align:middle line:84% that means that Aj intersect Bk is nonempty. 01:19:21.620 --> 01:19:24.230 align:middle line:84% And therefore that aj appearing there is going to be less than 01:19:24.230 --> 01:19:26.090 align:middle line:90% or equal to Bk. 01:19:26.090 --> 01:19:28.460 align:middle line:84% And whenever this is 0, well whatever's 01:19:28.460 --> 01:19:30.860 align:middle line:84% there is almost always less than or equal to Bk 01:19:30.860 --> 01:19:33.350 align:middle line:84% times the measure of Aj intersect Bk. 01:19:33.350 --> 01:19:37.850 align:middle line:84% So this is less than or equal to jk 01:19:37.850 --> 01:19:44.240 align:middle line:90% bk measure of Aj intersect Bk. 01:19:44.240 --> 01:19:51.790 align:middle line:90% 01:19:51.790 --> 01:19:54.970 align:middle line:84% So whenever this is-- so maybe that previous explanation 01:19:54.970 --> 01:19:55.540 align:middle line:90% was not good. 01:19:55.540 --> 01:20:00.730 align:middle line:84% Whenever this is nonempty I will always have aj less than 01:20:00.730 --> 01:20:01.510 align:middle line:90% or equal to bk. 01:20:01.510 --> 01:20:02.560 align:middle line:84% So I should have just said that I 01:20:02.560 --> 01:20:04.893 align:middle line:84% don't know what I was going on about measure zero stuff, 01:20:04.893 --> 01:20:06.130 align:middle line:90% but ignore that. 01:20:06.130 --> 01:20:09.850 align:middle line:84% Whenever this is nonempty, like I just said a minute ago, 01:20:09.850 --> 01:20:12.700 align:middle line:84% aj is less than or equal to bk, all right. 01:20:12.700 --> 01:20:19.360 align:middle line:90% 01:20:19.360 --> 01:20:21.100 align:middle line:90% And now we just reverse course. 01:20:21.100 --> 01:20:29.710 align:middle line:84% And this is equal to k equals 1 to m of bk measure of Bk 01:20:29.710 --> 01:20:34.450 align:middle line:84% because union over j's of these sets give me Bk. 01:20:34.450 --> 01:20:39.190 align:middle line:84% And this is equal to the integral of psi of E. 01:20:39.190 --> 01:20:44.350 align:middle line:84% And, again, I will leave 4 to you as a very simple exercise. 01:20:44.350 --> 01:20:50.130 align:middle line:90% 01:20:50.130 --> 01:20:54.190 align:middle line:90% OK, I'm about out of time. 01:20:54.190 --> 01:20:57.210 align:middle line:84% So what we've done is we've defined the Lebesgue 01:20:57.210 --> 01:20:58.950 align:middle line:90% integral of a simple function. 01:20:58.950 --> 01:21:01.660 align:middle line:90% 01:21:01.660 --> 01:21:07.690 align:middle line:84% And as that picture shows, I hope, or at least convinces you 01:21:07.690 --> 01:21:12.340 align:middle line:84% that if the simple function takes on that of a step 01:21:12.340 --> 01:21:16.540 align:middle line:84% function, meaning the Aj's are just intervals, 01:21:16.540 --> 01:21:19.930 align:middle line:84% then the Lebesgue integral of that step function 01:21:19.930 --> 01:21:22.300 align:middle line:84% will, in fact, be the area underneath phi. 01:21:22.300 --> 01:21:26.610 align:middle line:90% So it's, again, this is-- 01:21:26.610 --> 01:21:28.860 align:middle line:84% you can think of that as two ways, 01:21:28.860 --> 01:21:33.000 align:middle line:84% as the Lebesgue integral giving a theory of the area 01:21:33.000 --> 01:21:35.460 align:middle line:90% underneath the curve. 01:21:35.460 --> 01:21:42.160 align:middle line:84% You may also think of that as being the first indication 01:21:42.160 --> 01:21:44.740 align:middle line:84% that when I have a Riemann integral function, 01:21:44.740 --> 01:21:47.860 align:middle line:84% it will be also Lebesgue integral because if I have 01:21:47.860 --> 01:21:50.290 align:middle line:84% a step function like phi is in the picture, 01:21:50.290 --> 01:21:52.900 align:middle line:84% then that is a Riemann integral function. 01:21:52.900 --> 01:21:54.400 align:middle line:84% And the Riemann integral is the area 01:21:54.400 --> 01:21:59.290 align:middle line:84% underneath the curve, which also agrees with the definition 01:21:59.290 --> 01:22:01.040 align:middle line:90% of the Lebesgue integral. 01:22:01.040 --> 01:22:05.840 align:middle line:84% So that, like I said, should maybe 01:22:05.840 --> 01:22:11.750 align:middle line:84% indicate that the Lebesgue integral reduces to the Riemann 01:22:11.750 --> 01:22:15.770 align:middle line:84% integral whenever we're integrating a Riemann 01:22:15.770 --> 01:22:18.630 align:middle line:90% integrable function. 01:22:18.630 --> 01:22:23.870 align:middle line:84% And so next time we will define the integral 01:22:23.870 --> 01:22:30.980 align:middle line:84% of a non-negative measurable function using 01:22:30.980 --> 01:22:33.890 align:middle line:84% how we've defined the integral of simple functions, 01:22:33.890 --> 01:22:41.050 align:middle line:84% and prove some basic properties, including 01:22:41.050 --> 01:22:44.140 align:middle line:84% two of the main convergence theorems 01:22:44.140 --> 01:22:47.240 align:middle line:84% that go along with this theory of integration. 01:22:47.240 --> 01:22:49.230 align:middle line:90% So we'll stop there. 01:22:49.230 --> 01:22:55.000 align:middle line:90%