WEBVTT 00:00:00.000 --> 00:00:01.990 align:middle line:90% [SQUEAKING] 00:00:01.990 --> 00:00:02.490 align:middle line:90% 00:00:02.490 --> 00:00:03.984 align:middle line:90% [RUSTLING] 00:00:03.984 --> 00:00:04.980 align:middle line:90% [CLICKING] 00:00:04.980 --> 00:00:18.440 align:middle line:90% 00:00:18.440 --> 00:00:21.830 align:middle line:84% PROFESSOR: All right, so last time we 00:00:21.830 --> 00:00:24.080 align:middle line:84% defined the integral of a non-negative measurable 00:00:24.080 --> 00:00:27.800 align:middle line:90% function, the Lesbesguen rule. 00:00:27.800 --> 00:00:30.200 align:middle line:84% Now we are going to define the Lesbesguen 00:00:30.200 --> 00:00:33.660 align:middle line:84% rule for a general class, a more general class of functions. 00:00:33.660 --> 00:00:46.990 align:middle line:84% So Lesbesgue integrable functions. 00:00:46.990 --> 00:00:50.660 align:middle line:90% 00:00:50.660 --> 00:00:54.250 align:middle line:90% So what does this mean? 00:00:54.250 --> 00:00:57.360 align:middle line:90% 00:00:57.360 --> 00:01:07.010 align:middle line:84% Let E be a measurable subset of R, a function f, 00:01:07.010 --> 00:01:10.670 align:middle line:84% a measurable function f, from E to R. 00:01:10.670 --> 00:01:30.090 align:middle line:84% So its real value is Lesbesgue integrable over E. 00:01:30.090 --> 00:01:33.720 align:middle line:84% So I should say this is a measurable function-- 00:01:33.720 --> 00:01:42.960 align:middle line:84% is Lesbesgue integrable over E if the integral 00:01:42.960 --> 00:01:46.200 align:middle line:84% of the absolute value of f is finite. 00:01:46.200 --> 00:01:52.650 align:middle line:90% 00:01:52.650 --> 00:01:58.090 align:middle line:90% So a mark. 00:01:58.090 --> 00:02:01.510 align:middle line:90% 00:02:01.510 --> 00:02:04.210 align:middle line:84% So recall that we have the positive and negative parts 00:02:04.210 --> 00:02:06.220 align:middle line:90% of f. 00:02:06.220 --> 00:02:12.370 align:middle line:84% So we can write f equals f plus minus f minus, where these are 00:02:12.370 --> 00:02:14.470 align:middle line:84% the positive and negative parts of f, 00:02:14.470 --> 00:02:22.560 align:middle line:84% both non-negative functions so that the absolute value of f 00:02:22.560 --> 00:02:25.605 align:middle line:90% is equal to f plus plus f minus. 00:02:25.605 --> 00:02:33.270 align:middle line:90% 00:02:33.270 --> 00:02:35.310 align:middle line:84% And therefore, if I want to compute 00:02:35.310 --> 00:02:38.400 align:middle line:84% the integral of the absolute value, 00:02:38.400 --> 00:02:45.522 align:middle line:84% this is equal to the integral of the positive part 00:02:45.522 --> 00:02:46.480 align:middle line:90% plus the negative part. 00:02:46.480 --> 00:02:49.190 align:middle line:84% Remember, these are both non-negative functions. 00:02:49.190 --> 00:02:56.765 align:middle line:84% So let me just recall for you f plus is equal to max f0. 00:02:56.765 --> 00:02:58.930 align:middle line:90% f minus is equal to-- 00:02:58.930 --> 00:03:01.680 align:middle line:90% 00:03:01.680 --> 00:03:08.600 align:middle line:84% I think d for some reason, the f plus is always easy. 00:03:08.600 --> 00:03:11.670 align:middle line:90% The f minus is always-- 00:03:11.670 --> 00:03:13.830 align:middle line:90% max minus f0. 00:03:13.830 --> 00:03:20.980 align:middle line:90% 00:03:20.980 --> 00:03:25.450 align:middle line:84% So the integral of the absolute value 00:03:25.450 --> 00:03:28.420 align:middle line:84% is equal to the integral of f plus plus the integral of f 00:03:28.420 --> 00:03:29.080 align:middle line:90% minus. 00:03:29.080 --> 00:03:30.580 align:middle line:84% These are both defined because these 00:03:30.580 --> 00:03:33.070 align:middle line:84% are both non-negative measurable functions. 00:03:33.070 --> 00:03:38.260 align:middle line:90% 00:03:38.260 --> 00:03:40.330 align:middle line:84% So both non-negative because this 00:03:40.330 --> 00:03:43.390 align:middle line:84% is a max of two things, one of them involving 0. 00:03:43.390 --> 00:03:47.455 align:middle line:84% So that's always bigger than or equal to 0. 00:03:47.455 --> 00:03:49.580 align:middle line:84% So these are two non-negative measurable functions. 00:03:49.580 --> 00:03:50.850 align:middle line:90% These integrals exist. 00:03:50.850 --> 00:03:55.340 align:middle line:84% So this is finite if and only if both of these two things 00:03:55.340 --> 00:03:56.480 align:middle line:90% are finite. 00:03:56.480 --> 00:04:04.380 align:middle line:90% So thus f is-- 00:04:04.380 --> 00:04:06.510 align:middle line:84% instead of saying Lesbesgue integrable, 00:04:06.510 --> 00:04:09.240 align:middle line:84% I'm just going to say is integrable. 00:04:09.240 --> 00:04:13.680 align:middle line:84% This is equivalent to the functions 00:04:13.680 --> 00:04:18.887 align:middle line:84% f plus and f minus, the positive and negative parts of f, 00:04:18.887 --> 00:04:20.271 align:middle line:90% are integrable. 00:04:20.271 --> 00:04:26.280 align:middle line:90% 00:04:26.280 --> 00:04:33.570 align:middle line:84% All right, so with that remark there, 00:04:33.570 --> 00:04:37.220 align:middle line:84% so if f is a measurable function from E to a measurable set, 00:04:37.220 --> 00:04:41.940 align:middle line:90% the R is integrable. 00:04:41.940 --> 00:04:43.220 align:middle line:90% So "intbl." 00:04:43.220 --> 00:04:47.900 align:middle line:90% My short form is usually-- 00:04:47.900 --> 00:04:51.100 align:middle line:90% you can sound it out. 00:04:51.100 --> 00:05:07.600 align:middle line:84% If f is Lesbesgue integrable, then the Lesbesgue integral 00:05:07.600 --> 00:05:15.610 align:middle line:84% of f over E is defined to be the integral of f 00:05:15.610 --> 00:05:21.460 align:middle line:84% plus over E minus the integral of f minus. 00:05:21.460 --> 00:05:25.900 align:middle line:84% Now again, so this is a bit of new terminology. 00:05:25.900 --> 00:05:29.320 align:middle line:84% This is meaningful because I'm only defining the integral 00:05:29.320 --> 00:05:30.880 align:middle line:84% for integrable functions, meaning 00:05:30.880 --> 00:05:34.490 align:middle line:84% this is a finite number, this is a finite number. 00:05:34.490 --> 00:05:36.460 align:middle line:84% So I can always subtract two finite numbers. 00:05:36.460 --> 00:05:40.640 align:middle line:90% 00:05:40.640 --> 00:05:45.430 align:middle line:84% So this is the definition of being Lesbesgue integrable, 00:05:45.430 --> 00:05:47.040 align:middle line:90% the Lesbesgue integral. 00:05:47.040 --> 00:05:50.780 align:middle line:84% So what are some immediate properties 00:05:50.780 --> 00:05:54.050 align:middle line:84% of integrable functions and the integral? 00:05:54.050 --> 00:06:03.930 align:middle line:84% Suppose f and g from E to R are integrable. 00:06:03.930 --> 00:06:06.490 align:middle line:90% 00:06:06.490 --> 00:06:14.605 align:middle line:84% Then for all c in R, the integral of c times f-- 00:06:14.605 --> 00:06:25.050 align:middle line:84% no, I should say c times f is integrable. 00:06:25.050 --> 00:06:31.360 align:middle line:84% And the integral of c times f is equal to c 00:06:31.360 --> 00:06:32.740 align:middle line:90% times the integral of f. 00:06:32.740 --> 00:06:36.490 align:middle line:90% 00:06:36.490 --> 00:06:38.110 align:middle line:84% That's one simple property that just 00:06:38.110 --> 00:06:42.640 align:middle line:84% follows from the definition along with the linearity 00:06:42.640 --> 00:06:45.340 align:middle line:84% that we have for the integral of non-negative functions 00:06:45.340 --> 00:06:46.615 align:middle line:90% over non-negative scalars. 00:06:46.615 --> 00:06:49.910 align:middle line:90% 00:06:49.910 --> 00:06:54.770 align:middle line:90% f plus g is integrable. 00:06:54.770 --> 00:07:00.140 align:middle line:84% And the integral of f plus g over E 00:07:00.140 --> 00:07:09.920 align:middle line:84% is equal to the integral over E of f plus the integral of g. 00:07:09.920 --> 00:07:20.870 align:middle line:84% And again, just an analog of what 00:07:20.870 --> 00:07:24.870 align:middle line:84% we had for the integral of non-negative functions. 00:07:24.870 --> 00:07:35.910 align:middle line:84% If A and B are disjoint measurable sets, 00:07:35.910 --> 00:07:40.290 align:middle line:84% then integral of f over A union B 00:07:40.290 --> 00:07:42.990 align:middle line:84% is equal to the integral over A of f 00:07:42.990 --> 00:07:48.390 align:middle line:90% plus the integral over B of f. 00:07:48.390 --> 00:08:01.960 align:middle line:90% 00:08:01.960 --> 00:08:03.460 align:middle line:84% One I'm just going to leave to you. 00:08:03.460 --> 00:08:04.520 align:middle line:90% It's pretty clear. 00:08:04.520 --> 00:08:07.210 align:middle line:90% I just write c as-- 00:08:07.210 --> 00:08:09.610 align:middle line:84% if it's 0, then this follows immediately. 00:08:09.610 --> 00:08:11.440 align:middle line:84% If c is positive, then it doesn't 00:08:11.440 --> 00:08:13.540 align:middle line:84% change the positive and negative parts. 00:08:13.540 --> 00:08:16.180 align:middle line:84% The positive part of c times f is just 00:08:16.180 --> 00:08:19.258 align:middle line:90% c times the positive part of f. 00:08:19.258 --> 00:08:21.550 align:middle line:84% And the negative part is just c times the negative part 00:08:21.550 --> 00:08:26.480 align:middle line:84% of f is the negative part of c times f. 00:08:26.480 --> 00:08:28.910 align:middle line:84% But it flips them if c is negative. 00:08:28.910 --> 00:08:35.130 align:middle line:84% So you can just check that equality for those two cases. 00:08:35.130 --> 00:08:37.380 align:middle line:84% So let's move on to something more interesting, 00:08:37.380 --> 00:08:39.890 align:middle line:84% so for example, the fact that the integral 00:08:39.890 --> 00:08:43.820 align:middle line:90% is linear in f and g. 00:08:43.820 --> 00:08:48.560 align:middle line:84% So note that by the triangle inequality, I have 00:08:48.560 --> 00:08:53.960 align:middle line:84% that f plus g is less than or equal to f 00:08:53.960 --> 00:08:57.200 align:middle line:84% plus g, the absolute-- so the absolute value of f plus g 00:08:57.200 --> 00:08:59.210 align:middle line:84% is less than or equal to the absolute value of f 00:08:59.210 --> 00:09:00.920 align:middle line:90% plus the absolute value of g. 00:09:00.920 --> 00:09:04.358 align:middle line:84% And therefore, by what we know for the integral 00:09:04.358 --> 00:09:05.900 align:middle line:84% of non-negative measurable functions, 00:09:05.900 --> 00:09:10.400 align:middle line:84% the integral of f plus g is less than 00:09:10.400 --> 00:09:14.960 align:middle line:84% or equal to the integral of the absolute value of f 00:09:14.960 --> 00:09:17.120 align:middle line:84% plus the absolute value of g, which 00:09:17.120 --> 00:09:21.830 align:middle line:84% equals by linearity for non-negative measurable 00:09:21.830 --> 00:09:25.670 align:middle line:84% functions is equal to the sum of the integrals. 00:09:25.670 --> 00:09:27.680 align:middle line:90% Each of these is finite. 00:09:27.680 --> 00:09:29.210 align:middle line:84% So therefore, their sum is finite. 00:09:29.210 --> 00:09:32.190 align:middle line:90% 00:09:32.190 --> 00:09:39.380 align:middle line:84% So if I have two integrable functions, 00:09:39.380 --> 00:09:40.639 align:middle line:90% their sum is integrable. 00:09:40.639 --> 00:09:46.330 align:middle line:90% 00:09:46.330 --> 00:09:49.210 align:middle line:84% Now why is the integral of the sum 00:09:49.210 --> 00:09:50.770 align:middle line:84% equal to the sum of the integrals? 00:09:50.770 --> 00:09:56.320 align:middle line:84% Well, we have f plus g, splitting 00:09:56.320 --> 00:10:02.660 align:middle line:84% f and g into their positive and negative parts 00:10:02.660 --> 00:10:07.250 align:middle line:84% and writing f as f plus minus f minus. 00:10:07.250 --> 00:10:12.170 align:middle line:84% I can write f plus g as f plus plus g plus minus f 00:10:12.170 --> 00:10:14.600 align:middle line:90% minus plus g minus. 00:10:14.600 --> 00:10:18.740 align:middle line:84% Now this does not say that the positive part of f plus g 00:10:18.740 --> 00:10:22.130 align:middle line:84% is equal to this or that the negative part of f plus g 00:10:22.130 --> 00:10:23.000 align:middle line:90% is this. 00:10:23.000 --> 00:10:29.450 align:middle line:84% But it does say that the positive part 00:10:29.450 --> 00:10:37.620 align:middle line:84% of f plus g plus the sum of the negative parts is equal to-- 00:10:37.620 --> 00:10:42.150 align:middle line:90% 00:10:42.150 --> 00:10:45.240 align:middle line:84% just by splitting the left side into positive and negative 00:10:45.240 --> 00:10:47.430 align:middle line:90% parts, I get this identity. 00:10:47.430 --> 00:10:51.700 align:middle line:90% 00:10:51.700 --> 00:10:54.640 align:middle line:84% Now all of these are non-negative measurable 00:10:54.640 --> 00:10:56.552 align:middle line:90% functions. 00:10:56.552 --> 00:10:58.760 align:middle line:84% Everything appearing here are non-negative measurable 00:10:58.760 --> 00:10:59.690 align:middle line:90% functions. 00:10:59.690 --> 00:11:01.880 align:middle line:84% And therefore, the integral, which 00:11:01.880 --> 00:11:05.030 align:middle line:84% is linear for non-negative measurable functions, 00:11:05.030 --> 00:11:12.400 align:middle line:84% tells me that the integral of f plus g plus the integral of f 00:11:12.400 --> 00:11:21.870 align:middle line:84% minus plus g minus equals the integral of f plus plus g 00:11:21.870 --> 00:11:29.145 align:middle line:84% plus over E plus the integral over E plus g minus. 00:11:29.145 --> 00:11:32.130 align:middle line:90% 00:11:32.130 --> 00:11:37.830 align:middle line:90% Now what do I do is I rearrange. 00:11:37.830 --> 00:11:40.980 align:middle line:84% I bring this over to this side, this over to this side. 00:11:40.980 --> 00:11:43.800 align:middle line:84% And I can still use linearity because this is an integral 00:11:43.800 --> 00:11:45.900 align:middle line:90% of non-negative functions. 00:11:45.900 --> 00:11:50.430 align:middle line:84% I get that the integral of E over f 00:11:50.430 --> 00:11:57.330 align:middle line:84% plus g plus minus the integral of E over f plus g minus 00:11:57.330 --> 00:12:00.370 align:middle line:90% is equal to-- 00:12:00.370 --> 00:12:08.200 align:middle line:90% so let's just do this slowly. 00:12:08.200 --> 00:12:17.660 align:middle line:84% This is equal to this integral and this integral. 00:12:17.660 --> 00:12:21.910 align:middle line:84% So even though we didn't know that the positive part and-- 00:12:21.910 --> 00:12:23.960 align:middle line:84% equals the sum of the positive parts, 00:12:23.960 --> 00:12:26.200 align:middle line:84% so even though we don't know that the positive part 00:12:26.200 --> 00:12:27.850 align:middle line:84% of the sum is the sum of the positive parts 00:12:27.850 --> 00:12:29.558 align:middle line:84% and the same thing for the negative part, 00:12:29.558 --> 00:12:31.990 align:middle line:84% we still get the integrals equal. 00:12:31.990 --> 00:12:38.630 align:middle line:84% And this thing right here is just equal to, by definition, 00:12:38.630 --> 00:12:42.160 align:middle line:90% the integral of f plus g. 00:12:42.160 --> 00:12:44.020 align:middle line:90% And this thing over here-- 00:12:44.020 --> 00:12:47.950 align:middle line:84% again, if we use linearity for the integrals of non-negative 00:12:47.950 --> 00:13:01.040 align:middle line:84% functions, this is equal to f plus plus the integral over E 00:13:01.040 --> 00:13:02.500 align:middle line:90% of g plus, minus-- 00:13:02.500 --> 00:13:05.170 align:middle line:84% and again, just expanding this and then 00:13:05.170 --> 00:13:09.940 align:middle line:84% carrying through the minus, minus F minus. 00:13:09.940 --> 00:13:13.660 align:middle line:84% So you see here, all we're using is linearity for non-negative 00:13:13.660 --> 00:13:20.330 align:middle line:84% for a sum of two non-negative measurable functions. 00:13:20.330 --> 00:13:22.670 align:middle line:84% And this is just, again, by the definition 00:13:22.670 --> 00:13:28.780 align:middle line:84% of the integral the sum of the integrals. 00:13:28.780 --> 00:13:32.080 align:middle line:90% 00:13:32.080 --> 00:13:36.630 align:middle line:90% And so that was 2. 00:13:36.630 --> 00:13:37.967 align:middle line:90% What's the proof of 3? 00:13:37.967 --> 00:13:41.950 align:middle line:90% 00:13:41.950 --> 00:14:04.030 align:middle line:84% 3 from 2 and the fact that if I take f times the indicator 00:14:04.030 --> 00:14:07.660 align:middle line:84% function of A union B, this is equal to f times the indicator 00:14:07.660 --> 00:14:11.710 align:middle line:84% function of A plus f times the indicator function of B 00:14:11.710 --> 00:14:13.870 align:middle line:84% when A and B are two disjoint sets. 00:14:13.870 --> 00:14:19.860 align:middle line:90% 00:14:19.860 --> 00:14:25.400 align:middle line:84% So using what we know about for the integral 00:14:25.400 --> 00:14:27.440 align:middle line:84% of non-negative measurable functions, 00:14:27.440 --> 00:14:30.800 align:middle line:84% it follows that the integral of f over A union B 00:14:30.800 --> 00:14:33.410 align:middle line:84% is equal to the integral of this quantity 00:14:33.410 --> 00:14:38.030 align:middle line:84% here over E, which is equal to the sum of the integral 00:14:38.030 --> 00:14:41.280 align:middle line:84% of this sum here, which is equal to the sum of the integrals. 00:14:41.280 --> 00:14:43.640 align:middle line:84% And then going back, that's equal to the integral 00:14:43.640 --> 00:14:48.650 align:middle line:84% of f over A plus the integral of f over B. 00:14:48.650 --> 00:14:55.820 align:middle line:84% So again, it follows from linearity, this fact, 00:14:55.820 --> 00:14:57.560 align:middle line:84% and although I didn't write it down, 00:14:57.560 --> 00:15:01.160 align:middle line:84% the fact that even now for integrable functions, 00:15:01.160 --> 00:15:03.890 align:middle line:84% but that the integral of a subset 00:15:03.890 --> 00:15:06.597 align:middle line:84% is equal to the integral of the function times the indicator. 00:15:06.597 --> 00:15:08.180 align:middle line:84% The reason that's true is because it's 00:15:08.180 --> 00:15:10.970 align:middle line:84% true for non-negative measurable functions 00:15:10.970 --> 00:15:17.525 align:middle line:84% and simply from the definition of the Lesbesgue integral. 00:15:17.525 --> 00:15:27.240 align:middle line:90% 00:15:27.240 --> 00:15:30.180 align:middle line:84% All right, so some more properties of the integral. 00:15:30.180 --> 00:15:34.390 align:middle line:90% 00:15:34.390 --> 00:15:43.570 align:middle line:84% Suppose f and g from a measurable subset to R 00:15:43.570 --> 00:15:44.440 align:middle line:90% are integrable. 00:15:44.440 --> 00:15:49.640 align:middle line:90% 00:15:49.640 --> 00:15:54.140 align:middle line:90% Then the following holds. 00:15:54.140 --> 00:15:57.260 align:middle line:84% The integral-- I mean, the absolute value of the integral 00:15:57.260 --> 00:16:01.370 align:middle line:84% is less than or equal to the integral of the absolute value. 00:16:01.370 --> 00:16:05.620 align:middle line:90% 00:16:05.620 --> 00:16:18.510 align:middle line:84% If f equals g almost everywhere, then-- 00:16:18.510 --> 00:16:23.570 align:middle line:90% so let me-- so let's just-- 00:16:23.570 --> 00:16:26.210 align:middle line:84% I have no idea what I was going to say. 00:16:26.210 --> 00:16:30.670 align:middle line:84% I had to cut it down a little differently. 00:16:30.670 --> 00:16:33.010 align:middle line:90% So let's restart. 00:16:33.010 --> 00:16:36.640 align:middle line:84% Suppose f and g are two functions measurable. 00:16:36.640 --> 00:16:39.301 align:middle line:90% Need to write measurable. 00:16:39.301 --> 00:16:48.000 align:middle line:84% OK, back from the top, if f is integrable, 00:16:48.000 --> 00:16:51.810 align:middle line:84% then the absolute value of the integral of f 00:16:51.810 --> 00:16:55.995 align:middle line:84% is less than or equal to the integral of the absolute value. 00:16:55.995 --> 00:17:00.780 align:middle line:90% 00:17:00.780 --> 00:17:13.310 align:middle line:84% If g is integrable, and f equals g almost everywhere, 00:17:13.310 --> 00:17:18.170 align:middle line:90% then f is integrable. 00:17:18.170 --> 00:17:21.550 align:middle line:90% 00:17:21.550 --> 00:17:25.960 align:middle line:84% And the integral of f over E equals the integral 00:17:25.960 --> 00:17:31.540 align:middle line:84% of g, again, back to that philosophy from last lecture 00:17:31.540 --> 00:17:37.180 align:middle line:84% where I said if your conclusions are in terms of integrals, 00:17:37.180 --> 00:17:42.820 align:middle line:84% then usually the hypotheses can be stated only 00:17:42.820 --> 00:17:44.890 align:middle line:84% in terms of almost everywhere information. 00:17:44.890 --> 00:17:47.730 align:middle line:90% 00:17:47.730 --> 00:17:59.250 align:middle line:84% 3, if f and g our integrable, and f 00:17:59.250 --> 00:18:05.640 align:middle line:84% is less than or equal to g almost everywhere, 00:18:05.640 --> 00:18:08.790 align:middle line:84% then the integral of f over E is less than 00:18:08.790 --> 00:18:12.540 align:middle line:90% or equal to the integral of g. 00:18:12.540 --> 00:18:29.690 align:middle line:90% 00:18:29.690 --> 00:18:34.820 align:middle line:84% So one follows simply from the definition and the relationship 00:18:34.820 --> 00:18:36.872 align:middle line:84% between the absolute value of f, f, 00:18:36.872 --> 00:18:38.330 align:middle line:84% and the positive and negative part. 00:18:38.330 --> 00:18:45.970 align:middle line:84% So we have the absolute value of the integral of E of f over E 00:18:45.970 --> 00:18:49.240 align:middle line:84% of f is equal to, by definition, the integral 00:18:49.240 --> 00:18:55.660 align:middle line:84% of the positive part of E minus the negative part of E. 00:18:55.660 --> 00:18:57.530 align:middle line:84% Now these are two non-negative numbers. 00:18:57.530 --> 00:19:00.033 align:middle line:84% So by the triangle inequality, the absolute value 00:19:00.033 --> 00:19:02.200 align:middle line:84% of their difference is less than or equal to the sum 00:19:02.200 --> 00:19:05.920 align:middle line:84% of the absolute values, which are equal to themselves 00:19:05.920 --> 00:19:07.765 align:middle line:84% because they're non-negative numbers. 00:19:07.765 --> 00:19:10.390 align:middle line:90% 00:19:10.390 --> 00:19:13.030 align:middle line:90% And now the integral of-- 00:19:13.030 --> 00:19:15.790 align:middle line:84% or the sum of integrals is equal to the integral of the sum. 00:19:15.790 --> 00:19:19.620 align:middle line:90% 00:19:19.620 --> 00:19:24.840 align:middle line:84% And f plus the positive part plus the negative part of f 00:19:24.840 --> 00:19:29.340 align:middle line:84% is equal to the absolute value of f. 00:19:29.340 --> 00:19:31.780 align:middle line:84% So the triangle inequality-- or not-- 00:19:31.780 --> 00:19:33.280 align:middle line:84% I shouldn't say triangle inequality. 00:19:33.280 --> 00:19:38.250 align:middle line:84% But I mean, if you think of the integral as the sum, 00:19:38.250 --> 00:19:40.900 align:middle line:84% then this is a version of the triangle inequality. 00:19:40.900 --> 00:19:45.780 align:middle line:84% But the triangle inequality for Lesbesgue integrals holds. 00:19:45.780 --> 00:19:58.870 align:middle line:84% So 2, so we have f is less than or equal to-- 00:19:58.870 --> 00:20:01.150 align:middle line:84% so first off, absolute value of f 00:20:01.150 --> 00:20:04.090 align:middle line:84% is equal to the absolute value of g 00:20:04.090 --> 00:20:12.000 align:middle line:84% almost everywhere, which implies what 00:20:12.000 --> 00:20:16.062 align:middle line:84% we know from the integral of non-negative measurable 00:20:16.062 --> 00:20:18.270 align:middle line:84% functions, that if I have two non-negative measurable 00:20:18.270 --> 00:20:21.750 align:middle line:84% functions who equal each other almost everywhere, 00:20:21.750 --> 00:20:22.875 align:middle line:90% then their integrals agree. 00:20:22.875 --> 00:20:28.510 align:middle line:90% 00:20:28.510 --> 00:20:31.460 align:middle line:90% And that's fine. 00:20:31.460 --> 00:20:32.880 align:middle line:90% Thus, f is integrable. 00:20:32.880 --> 00:20:41.560 align:middle line:90% 00:20:41.560 --> 00:20:47.320 align:middle line:84% Now why does the integral of f over E equal the integral of g 00:20:47.320 --> 00:20:48.100 align:middle line:90% over E? 00:20:48.100 --> 00:20:52.490 align:middle line:90% 00:20:52.490 --> 00:20:58.360 align:middle line:84% So since f equals g almost everywhere, 00:20:58.360 --> 00:21:05.150 align:middle line:84% f minus g in absolute value is equal to 0 almost everywhere. 00:21:05.150 --> 00:21:12.630 align:middle line:84% And therefore, if I look at the absolute value 00:21:12.630 --> 00:21:15.300 align:middle line:84% of the difference of these two integrals, 00:21:15.300 --> 00:21:24.710 align:middle line:84% this is by linearity equal to the integral of the difference, 00:21:24.710 --> 00:21:29.410 align:middle line:84% which by the triangle inequality for Lesbesgue integrals 00:21:29.410 --> 00:21:33.670 align:middle line:84% is less than or equal to the integral of the absolute value. 00:21:33.670 --> 00:21:35.620 align:middle line:84% And by what we know for the integral 00:21:35.620 --> 00:21:37.360 align:middle line:84% of non-negative measurable functions, 00:21:37.360 --> 00:21:40.420 align:middle line:84% if I have a function which is 0 almost everywhere, then 00:21:40.420 --> 00:21:44.760 align:middle line:90% its integral is 0. 00:21:44.760 --> 00:21:49.350 align:middle line:84% And therefore, the absolute value of-- or the integral 00:21:49.350 --> 00:21:53.400 align:middle line:84% of f over E must equal the integral of g over E. 00:21:53.400 --> 00:21:54.420 align:middle line:90% So that proves 2. 00:21:54.420 --> 00:21:57.590 align:middle line:90% 00:21:57.590 --> 00:22:04.430 align:middle line:84% And 3 is-- again, so we're just using the stuff 00:22:04.430 --> 00:22:07.250 align:middle line:84% that we know from the integral of non-negative measurable 00:22:07.250 --> 00:22:08.060 align:middle line:90% functions. 00:22:08.060 --> 00:22:12.560 align:middle line:90% 00:22:12.560 --> 00:22:23.030 align:middle line:84% Find a function h of x to be g of x minus f of x and 0. 00:22:23.030 --> 00:22:24.020 align:middle line:90% Now when is it this? 00:22:24.020 --> 00:22:36.510 align:middle line:84% This is when if g of x is bigger than or equal to f of x, and 0 00:22:36.510 --> 00:22:37.450 align:middle line:90% otherwise. 00:22:37.450 --> 00:22:41.100 align:middle line:84% So this is a non-negative measurable function. 00:22:41.100 --> 00:22:44.640 align:middle line:84% It's equal to g of x minus f of x when g of x 00:22:44.640 --> 00:22:47.220 align:middle line:84% is bigger than or equal to f of x, and 0 otherwise. 00:22:47.220 --> 00:22:52.010 align:middle line:90% 00:22:52.010 --> 00:22:55.780 align:middle line:84% And h is a non-negative measurable function. 00:22:55.780 --> 00:23:00.460 align:middle line:84% h equals-- since g is bigger than or equal to f almost 00:23:00.460 --> 00:23:04.810 align:middle line:84% everywhere, h is equal to g minus f 00:23:04.810 --> 00:23:06.520 align:middle line:84% almost everywhere because this condition 00:23:06.520 --> 00:23:08.230 align:middle line:90% is satisfied almost everywhere. 00:23:08.230 --> 00:23:28.800 align:middle line:90% 00:23:28.800 --> 00:23:34.370 align:middle line:84% And therefore, I get that 0, which 00:23:34.370 --> 00:23:39.020 align:middle line:84% is less than or equal to the integral over E 00:23:39.020 --> 00:23:41.900 align:middle line:90% of the positive part of h-- 00:23:41.900 --> 00:23:43.970 align:middle line:84% because this is just a non-negative function, 00:23:43.970 --> 00:23:46.178 align:middle line:84% we know what its Lesbesgue integral is defined to be. 00:23:46.178 --> 00:23:51.060 align:middle line:84% It's just given by the previous lecture. 00:23:51.060 --> 00:23:53.480 align:middle line:84% So it has to be a non-negative number, which 00:23:53.480 --> 00:23:57.125 align:middle line:84% because h is non-negative, this is equal to h. 00:23:57.125 --> 00:23:58.715 align:middle line:90% h plus is equal to h. 00:23:58.715 --> 00:24:01.370 align:middle line:90% 00:24:01.370 --> 00:24:08.560 align:middle line:84% But what we proved a minute ago, since these two functions, 00:24:08.560 --> 00:24:21.700 align:middle line:84% integrable function, so I should say why. 00:24:21.700 --> 00:24:24.710 align:middle line:84% So why am I also asserting that h is an integrable function? 00:24:24.710 --> 00:24:27.430 align:middle line:84% Well, the integral of h is equal to the integral 00:24:27.430 --> 00:24:29.830 align:middle line:84% of g minus f, which, if I want to put 00:24:29.830 --> 00:24:32.710 align:middle line:84% absolute values on things, the integral of the absolute value 00:24:32.710 --> 00:24:38.980 align:middle line:84% of h is equal to the difference of two integral functions, 00:24:38.980 --> 00:24:41.080 align:middle line:84% which is integrable almost Everywhere 00:24:41.080 --> 00:24:43.630 align:middle line:84% and by what we did for part 2, then 00:24:43.630 --> 00:24:47.720 align:middle line:90% that tells me h is integrable. 00:24:47.720 --> 00:24:54.030 align:middle line:84% So the integral of H, so by 2, must 00:24:54.030 --> 00:25:00.810 align:middle line:84% be equal to the integral of g minus f, which by the linearity 00:25:00.810 --> 00:25:03.570 align:middle line:84% that we proved in the theorem before is 00:25:03.570 --> 00:25:11.670 align:middle line:84% equal to the integral of g minus the integral of f. 00:25:11.670 --> 00:25:14.400 align:middle line:84% And you remember, we started all the way at the end of this 00:25:14.400 --> 00:25:17.190 align:middle line:84% or at the beginning of this with 0 is less than or equal to. 00:25:17.190 --> 00:25:19.170 align:middle line:84% So the integral of g will be bigger than 00:25:19.170 --> 00:25:20.460 align:middle line:90% or equal to the integral of f. 00:25:20.460 --> 00:25:26.400 align:middle line:90% 00:25:26.400 --> 00:25:35.910 align:middle line:84% So now the final most useful convergence theorem one 00:25:35.910 --> 00:25:38.655 align:middle line:84% encounters in Lesbesgue integration 00:25:38.655 --> 00:25:41.790 align:middle line:84% or in integration theory is Lesbesgue's 00:25:41.790 --> 00:25:43.098 align:middle line:90% dominated convergence theorem. 00:25:43.098 --> 00:25:45.390 align:middle line:84% Or I'll just call it the dominated convergence theorem. 00:25:45.390 --> 00:26:08.248 align:middle line:90% 00:26:08.248 --> 00:26:09.415 align:middle line:90% Let me make a small comment. 00:26:09.415 --> 00:26:12.560 align:middle line:90% 00:26:12.560 --> 00:26:14.990 align:middle line:84% Well, let me pause and make a small comment right here. 00:26:14.990 --> 00:26:21.440 align:middle line:84% What functions are Lesbesgue integrable first off? 00:26:21.440 --> 00:26:23.090 align:middle line:84% What are some examples of functions 00:26:23.090 --> 00:26:25.190 align:middle line:90% that are Lesbesgue integrable? 00:26:25.190 --> 00:26:29.251 align:middle line:90% 00:26:29.251 --> 00:26:31.375 align:middle line:90% Did I-- let's see. 00:26:31.375 --> 00:26:33.127 align:middle line:90% Did I skip that? 00:26:33.127 --> 00:26:34.210 align:middle line:90% Oh, I had it at one point. 00:26:34.210 --> 00:26:39.010 align:middle line:90% 00:26:39.010 --> 00:26:42.420 align:middle line:84% So what sets have finite measure? 00:26:42.420 --> 00:26:45.110 align:middle line:90% 00:26:45.110 --> 00:26:47.465 align:middle line:90% We know that the set-- 00:26:47.465 --> 00:26:50.270 align:middle line:84% the measure of intervals is the length of the intervals, right? 00:26:50.270 --> 00:26:56.690 align:middle line:84% So any sets that are contained in a large interval are-- 00:26:56.690 --> 00:26:59.960 align:middle line:84% any measurable subsets that are contained in a large interval 00:26:59.960 --> 00:27:01.010 align:middle line:90% have finite measures. 00:27:01.010 --> 00:27:04.705 align:middle line:84% So compact sets, which we know are measurable, because they're 00:27:04.705 --> 00:27:08.340 align:middle line:90% Borel sets, are-- 00:27:08.340 --> 00:27:11.474 align:middle line:90% have finite measure. 00:27:11.474 --> 00:27:15.960 align:middle line:90% So why am I saying that? 00:27:15.960 --> 00:27:21.750 align:middle line:84% If I have simple functions which are nonzero only on sets that 00:27:21.750 --> 00:27:26.970 align:middle line:84% have finite measure, then that will be an integrable function, 00:27:26.970 --> 00:27:33.970 align:middle line:84% because go back to the definition of how 00:27:33.970 --> 00:27:36.040 align:middle line:84% one integrates a simple function, right? 00:27:36.040 --> 00:27:40.320 align:middle line:84% It's the sum of the coefficients times the measure 00:27:40.320 --> 00:27:43.300 align:middle line:84% of the sets, where the-- it takes that coefficient. 00:27:43.300 --> 00:27:47.880 align:middle line:84% So if I'm only nonzero on a set of-- 00:27:47.880 --> 00:27:51.480 align:middle line:84% with the convention that 0 times infinity equals 0 00:27:51.480 --> 00:27:57.480 align:middle line:84% so that it's the measure of 0 times the measure of where 00:27:57.480 --> 00:28:04.060 align:middle line:84% the function is 0 is equal to 0, then those simple functions 00:28:04.060 --> 00:28:06.970 align:middle line:84% that are nonzero only on sets of finite measure 00:28:06.970 --> 00:28:08.560 align:middle line:90% will be integral functions. 00:28:08.560 --> 00:28:13.880 align:middle line:84% Now what about continuous functions? 00:28:13.880 --> 00:28:16.722 align:middle line:90% So in fact, let me-- 00:28:16.722 --> 00:28:18.430 align:middle line:84% I'm not going to prove this because we're 00:28:18.430 --> 00:28:21.620 align:middle line:84% going to prove something much stronger in a minute. 00:28:21.620 --> 00:28:24.590 align:middle line:84% So what about continuous functions 00:28:24.590 --> 00:28:27.320 align:middle line:84% on a closed and bounded interval, say a-- 00:28:27.320 --> 00:28:28.670 align:middle line:90% so a continuous function on AB? 00:28:28.670 --> 00:28:31.310 align:middle line:90% 00:28:31.310 --> 00:28:34.070 align:middle line:84% Let's just talk our way through why those functions 00:28:34.070 --> 00:28:35.880 align:middle line:90% are integrable on those sets. 00:28:35.880 --> 00:28:38.160 align:middle line:84% So a continuous function on an interval AB-- 00:28:38.160 --> 00:28:40.950 align:middle line:90% 00:28:40.950 --> 00:28:43.260 align:middle line:84% if a function is continuous, then its absolute value 00:28:43.260 --> 00:28:45.060 align:middle line:90% is continuous on AB. 00:28:45.060 --> 00:28:46.560 align:middle line:84% And therefore, the absolute value 00:28:46.560 --> 00:28:49.680 align:middle line:84% must be bounded by some constant. 00:28:49.680 --> 00:28:52.530 align:middle line:84% A continuous function attains a minimum and maximum 00:28:52.530 --> 00:28:54.550 align:middle line:84% on a closed and bounded interval. 00:28:54.550 --> 00:28:57.270 align:middle line:84% So the absolute value of a continuous function 00:28:57.270 --> 00:29:00.225 align:middle line:84% on a closed and bounded interval is bounded by some constant. 00:29:00.225 --> 00:29:03.020 align:middle line:90% 00:29:03.020 --> 00:29:06.650 align:middle line:84% And therefore, by monotonicity of Lesbesgue integrals, 00:29:06.650 --> 00:29:10.640 align:middle line:84% the integral of the absolute value of f over the interval AB 00:29:10.640 --> 00:29:13.010 align:middle line:84% will be less than or equal to some constant times 00:29:13.010 --> 00:29:15.930 align:middle line:90% the interval over the-- 00:29:15.930 --> 00:29:20.610 align:middle line:84% or times-- will be less than or equal to the integral 00:29:20.610 --> 00:29:22.680 align:middle line:90% of the constant over AB. 00:29:22.680 --> 00:29:24.570 align:middle line:84% And the integral of a constant is 00:29:24.570 --> 00:29:28.050 align:middle line:84% just equal to the constant times a measure of the set. 00:29:28.050 --> 00:29:31.830 align:middle line:84% Just by how we define the integral of simple functions, 00:29:31.830 --> 00:29:36.960 align:middle line:84% a constant is just the simplest of simple functions, which 00:29:36.960 --> 00:29:41.130 align:middle line:84% would be the constant bounding above the absolute value 00:29:41.130 --> 00:29:45.180 align:middle line:84% times the measure of the closed and bounded interval AB, which 00:29:45.180 --> 00:29:46.500 align:middle line:90% is B minus a 00:29:46.500 --> 00:29:49.990 align:middle line:84% So the Lesbesgue integral of a continuous function-- 00:29:49.990 --> 00:29:52.290 align:middle line:90% so what was all that for? 00:29:52.290 --> 00:29:54.600 align:middle line:84% To say that a continuous function 00:29:54.600 --> 00:29:58.465 align:middle line:84% on a closed and bounded interval is a Lesbesgue integrable. 00:29:58.465 --> 00:30:00.090 align:middle line:84% Now we're about to show something right 00:30:00.090 --> 00:30:02.850 align:middle line:84% after I prove the state and prove the dominated convergence 00:30:02.850 --> 00:30:06.240 align:middle line:84% theorem, which is much stronger than that, 00:30:06.240 --> 00:30:07.982 align:middle line:84% is that in fact, the Lesbesgue integral 00:30:07.982 --> 00:30:10.440 align:middle line:84% of a continuous function on the closed and bounded interval 00:30:10.440 --> 00:30:12.730 align:middle line:90% equals its Riemann integral. 00:30:12.730 --> 00:30:16.050 align:middle line:84% So in fact, for continuous functions, 00:30:16.050 --> 00:30:18.330 align:middle line:84% you already know how to compute Lesbesgue integrals. 00:30:18.330 --> 00:30:21.310 align:middle line:84% They're equal to the Riemann integrals. 00:30:21.310 --> 00:30:24.285 align:middle line:84% So back to the dominated convergence theorem. 00:30:24.285 --> 00:30:27.030 align:middle line:90% 00:30:27.030 --> 00:30:44.870 align:middle line:84% Let g be an integrable function over E. f 00:30:44.870 --> 00:30:56.860 align:middle line:84% in a sequence of measurable functions such that-- 00:30:56.860 --> 00:31:02.360 align:middle line:90% 00:31:02.360 --> 00:31:05.030 align:middle line:90% two things. 00:31:05.030 --> 00:31:10.045 align:middle line:84% One, for all n, the absolute value of fn 00:31:10.045 --> 00:31:11.170 align:middle line:90% is less than or equal to g. 00:31:11.170 --> 00:31:12.190 align:middle line:90% So g is non-negative. 00:31:12.190 --> 00:31:14.216 align:middle line:84% I guess I should have put here non-negative. 00:31:14.216 --> 00:31:20.080 align:middle line:90% 00:31:20.080 --> 00:31:37.170 align:middle line:84% And almost everywhere, there exists an f from E 00:31:37.170 --> 00:31:46.130 align:middle line:84% to R such that fn converges to f pointwise 00:31:46.130 --> 00:31:54.520 align:middle line:84% is almost everywhere, so meaning fn of x converges to f 00:31:54.520 --> 00:32:05.330 align:middle line:90% of x for almost every x in E. 00:32:05.330 --> 00:32:10.520 align:middle line:84% So these, the sequence of measurable functions, 00:32:10.520 --> 00:32:11.880 align:middle line:90% converge to-- 00:32:11.880 --> 00:32:14.000 align:middle line:84% pointwise to this function almost everywhere. 00:32:14.000 --> 00:32:17.970 align:middle line:84% And they're all dominated by an integrable function. 00:32:17.970 --> 00:32:29.660 align:middle line:84% Then the conclusion is that the limit of the integrals 00:32:29.660 --> 00:32:33.892 align:middle line:84% equals the integral of the limits 00:32:33.892 --> 00:32:35.100 align:middle line:90% or the integral of the limit. 00:32:35.100 --> 00:32:45.210 align:middle line:90% 00:32:45.210 --> 00:32:54.470 align:middle line:84% So this is a very useful and powerful theorem 00:32:54.470 --> 00:32:55.550 align:middle line:90% in integration. 00:32:55.550 --> 00:33:04.050 align:middle line:84% It's way stronger than you can-- than anything one can really 00:33:04.050 --> 00:33:09.780 align:middle line:84% say in Riemann integration, as far as convergence theorems go. 00:33:09.780 --> 00:33:12.510 align:middle line:84% Riemann integration requires always 00:33:12.510 --> 00:33:15.480 align:middle line:84% in some form, some form of uniform convergence here. 00:33:15.480 --> 00:33:18.030 align:middle line:84% All we have is pointwise convergence almost everywhere, 00:33:18.030 --> 00:33:18.810 align:middle line:90% right? 00:33:18.810 --> 00:33:21.890 align:middle line:84% And the second requirement-- so remember, 00:33:21.890 --> 00:33:24.390 align:middle line:84% the monotone convergence theorem also required the functions 00:33:24.390 --> 00:33:27.860 align:middle line:90% to be increasing. 00:33:27.860 --> 00:33:31.640 align:middle line:84% So here, now we're dealing not just 00:33:31.640 --> 00:33:34.310 align:middle line:84% with non-negative functions, just arbitrary measurable 00:33:34.310 --> 00:33:36.860 align:middle line:90% functions. 00:33:36.860 --> 00:33:53.010 align:middle line:84% Then I should say real valued measurable functions 00:33:53.010 --> 00:33:54.870 align:middle line:84% because our measurable functions could also 00:33:54.870 --> 00:33:55.890 align:middle line:90% be extended real value. 00:33:55.890 --> 00:33:57.520 align:middle line:84% But I wanted this to be real value. 00:33:57.520 --> 00:34:00.360 align:middle line:84% So all we need on top of pointwise convergence 00:34:00.360 --> 00:34:03.000 align:middle line:84% almost everywhere is just for these 00:34:03.000 --> 00:34:08.010 align:middle line:84% to be bounded above by some integrable function, some fixed 00:34:08.010 --> 00:34:09.190 align:middle line:90% integrable function. 00:34:09.190 --> 00:34:12.030 align:middle line:84% Then we conclude that the limit of the integrals 00:34:12.030 --> 00:34:15.750 align:middle line:84% is equal to the integral of the limit, which 00:34:15.750 --> 00:34:21.120 align:middle line:84% is an extremely powerful and useful theorem in analysis. 00:34:21.120 --> 00:34:25.820 align:middle line:90% 00:34:25.820 --> 00:34:28.070 align:middle line:84% So we'll prove it using Fatou's lemma. 00:34:28.070 --> 00:34:34.900 align:middle line:84% But first, again, the conclusions 00:34:34.900 --> 00:34:36.370 align:middle line:90% are about integrals. 00:34:36.370 --> 00:34:38.679 align:middle line:84% And I'm making almost everywhere statements 00:34:38.679 --> 00:34:42.190 align:middle line:84% in the assumptions, which I said you could always basically do. 00:34:42.190 --> 00:34:44.560 align:middle line:84% Let me briefly reduce ourselves to the case 00:34:44.560 --> 00:34:47.920 align:middle line:84% that these two things hold almost everywhere. 00:34:47.920 --> 00:34:49.300 align:middle line:90% I mean everywhere. 00:34:49.300 --> 00:34:54.429 align:middle line:84% Since for all n, fn is less than or equal 00:34:54.429 --> 00:35:02.260 align:middle line:84% to g almost everywhere, this implies that for all n, 00:35:02.260 --> 00:35:08.190 align:middle line:90% fn is integrable. 00:35:08.190 --> 00:35:14.460 align:middle line:90% 00:35:14.460 --> 00:35:23.310 align:middle line:84% So moreover, fn converges to f almost everywhere 00:35:23.310 --> 00:35:26.490 align:middle line:84% implies a couple of things, that-- 00:35:26.490 --> 00:35:29.520 align:middle line:84% remember, the pointwise even almost everywhere convergence 00:35:29.520 --> 00:35:32.620 align:middle line:84% of measurable functions is a measurable function. 00:35:32.620 --> 00:35:37.380 align:middle line:90% So f is measurable. 00:35:37.380 --> 00:35:51.000 align:middle line:84% And so f is less than or equal to g almost everywhere, which 00:35:51.000 --> 00:35:59.059 align:middle line:90% implies that f is integrable. 00:35:59.059 --> 00:36:04.110 align:middle line:90% 00:36:04.110 --> 00:36:20.060 align:middle line:84% So since changing fn for each n-- 00:36:20.060 --> 00:36:22.250 align:middle line:84% or I should say-- let me say this. 00:36:22.250 --> 00:36:29.910 align:middle line:90% Since changing f on a-- 00:36:29.910 --> 00:36:34.966 align:middle line:90% 00:36:34.966 --> 00:36:35.674 align:middle line:90% what am I saying? 00:36:35.674 --> 00:36:50.180 align:middle line:84% Since changing f and fn for all n on a set of measure 00:36:50.180 --> 00:37:05.015 align:middle line:90% 0 does not affect the integrals. 00:37:05.015 --> 00:37:06.390 align:middle line:84% In the end, our conclusion should 00:37:06.390 --> 00:37:10.500 align:middle line:84% be about the limit of the integral of the sequence equals 00:37:10.500 --> 00:37:14.250 align:middle line:90% the integral of f. 00:37:14.250 --> 00:37:18.660 align:middle line:84% And since if I change fn and f for each n 00:37:18.660 --> 00:37:23.940 align:middle line:84% on a set of measure 0, doesn't change the integrals, right? 00:37:23.940 --> 00:37:27.040 align:middle line:84% We can assume that those two assumptions-- 00:37:27.040 --> 00:37:28.650 align:middle line:90% we can assume that these two-- 00:37:28.650 --> 00:37:37.880 align:middle line:90% 00:37:37.880 --> 00:37:47.980 align:middle line:84% we can assume that these two assumptions hold everywhere. 00:37:47.980 --> 00:37:59.810 align:middle line:90% 00:37:59.810 --> 00:38:20.290 align:middle line:90% And these hold everywhere on E. 00:38:20.290 --> 00:38:22.210 align:middle line:84% So first off, I want to note-- so now let's 00:38:22.210 --> 00:38:23.560 align:middle line:90% actually get to the proof. 00:38:23.560 --> 00:38:25.310 align:middle line:84% All right, maybe you need to take a second 00:38:25.310 --> 00:38:29.300 align:middle line:90% to think about what I said here. 00:38:29.300 --> 00:38:33.010 align:middle line:84% But the point is that because the conclusions are 00:38:33.010 --> 00:38:35.770 align:middle line:84% in terms of integrals, and although I stated it 00:38:35.770 --> 00:38:41.110 align:middle line:84% for almost everywhere, I can fiddle 00:38:41.110 --> 00:38:46.240 align:middle line:84% with fn and f for each n on a fixed set of measure 0 00:38:46.240 --> 00:38:49.310 align:middle line:90% without affecting the integrals. 00:38:49.310 --> 00:38:52.870 align:middle line:84% So if I want to prove this, I can change them 00:38:52.870 --> 00:38:56.320 align:middle line:84% on a set of measure 0 without affecting the integrals. 00:38:56.320 --> 00:39:00.040 align:middle line:84% And on the set of measures 0, I make it so that all of the fns 00:39:00.040 --> 00:39:00.670 align:middle line:90% equal f. 00:39:00.670 --> 00:39:04.330 align:middle line:84% And so then I have this convergence everywhere. 00:39:04.330 --> 00:39:07.540 align:middle line:84% And I change f to be 0 on that set as well so that I will 00:39:07.540 --> 00:39:09.490 align:middle line:90% have this everywhere as well. 00:39:09.490 --> 00:39:12.780 align:middle line:90% 00:39:12.780 --> 00:39:16.680 align:middle line:84% But so if you need a second, if you'd like, 00:39:16.680 --> 00:39:19.380 align:middle line:84% just imagine that I erased that "almost everywhere" 00:39:19.380 --> 00:39:20.880 align:middle line:84% so that the statement of the theorem 00:39:20.880 --> 00:39:22.650 align:middle line:84% has these things holding everywhere. 00:39:22.650 --> 00:39:26.802 align:middle line:84% And then think a little bit about why 00:39:26.802 --> 00:39:28.260 align:middle line:84% I don't need it to hold everywhere, 00:39:28.260 --> 00:39:29.930 align:middle line:90% just almost everywhere. 00:39:29.930 --> 00:39:33.010 align:middle line:90% 00:39:33.010 --> 00:39:36.270 align:middle line:84% So we assume for all n, fn is less than or equal to g. 00:39:36.270 --> 00:39:40.160 align:middle line:84% There exists an f so that I have that. 00:39:40.160 --> 00:39:53.030 align:middle line:84% So note, for all n, the integral of f sub n is less than 00:39:53.030 --> 00:39:58.380 align:middle line:84% or equal to the integral of the absolute value, 00:39:58.380 --> 00:40:06.290 align:middle line:84% which is less than or equal to the integral of g, 00:40:06.290 --> 00:40:13.100 align:middle line:84% which implies that the sequence formed by the integrals 00:40:13.100 --> 00:40:16.970 align:middle line:84% of these guys-- so this is a sequence of real numbers-- 00:40:16.970 --> 00:40:25.760 align:middle line:84% is a bounded sequence of real numbers. 00:40:25.760 --> 00:40:28.470 align:middle line:90% 00:40:28.470 --> 00:40:31.560 align:middle line:84% So it has a limsup and a liminf, right? 00:40:31.560 --> 00:40:34.740 align:middle line:84% Now remember from your first analysis 00:40:34.740 --> 00:40:42.990 align:middle line:84% course, whatever it was, that the limit of a bounded sequence 00:40:42.990 --> 00:40:46.320 align:middle line:84% is equal to L if and only if the liminf and limsup equal 00:40:46.320 --> 00:40:48.900 align:middle line:84% each other, and they equal L. So what we're going to do 00:40:48.900 --> 00:40:51.870 align:middle line:84% is we're going to show that the liminf and the limsup 00:40:51.870 --> 00:40:54.510 align:middle line:84% of this sequence of numbers are equal. 00:40:54.510 --> 00:40:56.200 align:middle line:84% And they equal the integral of f. 00:40:56.200 --> 00:40:59.155 align:middle line:90% 00:40:59.155 --> 00:41:01.490 align:middle line:84% And to do that, we're going to use Fatou's lemma. 00:41:01.490 --> 00:41:11.105 align:middle line:84% So since g plus or minus f sub n is less-- 00:41:11.105 --> 00:41:15.085 align:middle line:84% or is bigger than or equal to 0 because the absolute value 00:41:15.085 --> 00:41:16.960 align:middle line:84% of f sub n is always less than or equal to g. 00:41:16.960 --> 00:41:19.120 align:middle line:84% So g plus or minus f sub n is always 00:41:19.120 --> 00:41:20.590 align:middle line:90% bigger than or equal to 0. 00:41:20.590 --> 00:41:29.040 align:middle line:84% I can now apply Fatou's lemma, which tells me 00:41:29.040 --> 00:41:32.310 align:middle line:84% that for a sequence of non-negative measurable 00:41:32.310 --> 00:41:35.340 align:middle line:84% functions, the integral of the liminf is less than 00:41:35.340 --> 00:41:37.200 align:middle line:84% or equal to the liminf of the integrals. 00:41:37.200 --> 00:41:41.780 align:middle line:90% 00:41:41.780 --> 00:41:53.510 align:middle line:84% So again, so the fns are converging to f. 00:41:53.510 --> 00:41:56.090 align:middle line:90% So f of x-- 00:41:56.090 --> 00:42:02.270 align:middle line:84% fn of x for all-- so fix x in E, fn of x converges to f of x. 00:42:02.270 --> 00:42:06.650 align:middle line:84% Therefore, liminf of fn of x equals f of x. 00:42:06.650 --> 00:42:13.710 align:middle line:84% So the liminf of g minus f equals fn. 00:42:13.710 --> 00:42:17.820 align:middle line:84% Sorry, equals g minus f because fns 00:42:17.820 --> 00:42:21.180 align:middle line:84% converge to f, again, for each pointwise. 00:42:21.180 --> 00:42:28.580 align:middle line:90% 00:42:28.580 --> 00:42:29.840 align:middle line:90% Let's do a minus here. 00:42:29.840 --> 00:42:32.590 align:middle line:90% 00:42:32.590 --> 00:42:37.570 align:middle line:84% And by Fatou's lemma, this is less than or equal to liminf 00:42:37.570 --> 00:42:40.720 align:middle line:84% as n goes to infinity of the integral of E-- 00:42:40.720 --> 00:42:43.930 align:middle line:90% over E of g minus f sub n. 00:42:43.930 --> 00:42:48.330 align:middle line:84% Now this is equal to the integral. 00:42:48.330 --> 00:42:51.810 align:middle line:84% Using linearity, this is equal to the integral of g 00:42:51.810 --> 00:42:55.590 align:middle line:90% minus the integral of f sub n. 00:42:55.590 --> 00:42:57.900 align:middle line:84% When I take the liminf of that and carry it through, 00:42:57.900 --> 00:43:03.080 align:middle line:84% I get the liminf when it hits a minus turns into a limsup. 00:43:03.080 --> 00:43:11.910 align:middle line:90% 00:43:11.910 --> 00:43:14.340 align:middle line:84% So whenever I have a bounded sequence of numbers, 00:43:14.340 --> 00:43:18.000 align:middle line:84% the liminf minus that sequence of numbers 00:43:18.000 --> 00:43:22.300 align:middle line:84% equals minus the limsup of that sequence of numbers. 00:43:22.300 --> 00:43:31.040 align:middle line:84% So here I'm using liminf of minus An, n being-- 00:43:31.040 --> 00:43:36.170 align:middle line:90% equals minus limsup of An. 00:43:36.170 --> 00:43:44.080 align:middle line:90% 00:43:44.080 --> 00:43:54.280 align:middle line:84% And similarly, I get that the integral of g plus f 00:43:54.280 --> 00:43:58.540 align:middle line:84% is less than or equal to just by-- 00:43:58.540 --> 00:44:00.490 align:middle line:84% now if I choose lemma again, but now I 00:44:00.490 --> 00:44:01.960 align:middle line:84% don't have to switch minuses, this 00:44:01.960 --> 00:44:08.260 align:middle line:84% is less than or equal to the integral of g plus liminf 00:44:08.260 --> 00:44:13.300 align:middle line:84% as n goes to infinity of the integral of fn of E. 00:44:13.300 --> 00:44:16.840 align:middle line:90% So I have two inequalities. 00:44:16.840 --> 00:44:25.400 align:middle line:90% I have that. 00:44:25.400 --> 00:44:28.370 align:middle line:84% And I should say it's less than or equal to, so maybe 00:44:28.370 --> 00:44:31.220 align:middle line:90% include a little bit of that. 00:44:31.220 --> 00:44:34.640 align:middle line:84% And then I also have this inequality here. 00:44:34.640 --> 00:44:45.390 align:middle line:90% 00:44:45.390 --> 00:44:49.110 align:middle line:84% Now all of these quantities that I've written down, 00:44:49.110 --> 00:44:50.520 align:middle line:90% these are all finite numbers. 00:44:50.520 --> 00:44:51.670 align:middle line:90% This is one reason why. 00:44:51.670 --> 00:44:54.540 align:middle line:84% So this is the limsup of this bounded sequence. 00:44:54.540 --> 00:44:55.360 align:middle line:90% That's a number. 00:44:55.360 --> 00:44:56.805 align:middle line:90% This is a number. 00:44:56.805 --> 00:44:57.700 align:middle line:90% That's a number. 00:44:57.700 --> 00:45:00.630 align:middle line:84% So I can subtract and move them to either side 00:45:00.630 --> 00:45:02.550 align:middle line:90% of this inequality. 00:45:02.550 --> 00:45:04.050 align:middle line:84% So there's no funny business going 00:45:04.050 --> 00:45:06.300 align:middle line:90% on with subtracting infinities. 00:45:06.300 --> 00:45:08.530 align:middle line:90% This is all on the level. 00:45:08.530 --> 00:45:12.240 align:middle line:84% So moving the limsup over and subtracting over 00:45:12.240 --> 00:45:18.930 align:middle line:84% the integral of g minus f, I get that the limsup as n 00:45:18.930 --> 00:45:24.420 align:middle line:84% goes to infinity of f sub n is less than 00:45:24.420 --> 00:45:33.440 align:middle line:84% or equal to the integral of g minus f. 00:45:33.440 --> 00:45:37.940 align:middle line:84% And now by linearity, that's equal to the integral of f. 00:45:37.940 --> 00:45:42.230 align:middle line:84% And this is also equal to, by linearity-- 00:45:42.230 --> 00:45:51.260 align:middle line:84% let's see-- minus the integral of g. 00:45:51.260 --> 00:45:57.950 align:middle line:84% And by the second yellow box here, this 00:45:57.950 --> 00:46:00.170 align:middle line:84% is less than or equal to the integral of g 00:46:00.170 --> 00:46:06.320 align:middle line:84% plus the liminf of the fn, so liminf 00:46:06.320 --> 00:46:07.655 align:middle line:90% of the integrals of the fns. 00:46:07.655 --> 00:46:10.890 align:middle line:90% 00:46:10.890 --> 00:46:13.680 align:middle line:90% So what do I have? 00:46:13.680 --> 00:46:20.220 align:middle line:84% I have the limsup is less than or equal to the integral of f, 00:46:20.220 --> 00:46:23.310 align:middle line:84% is less than or equal to the liminf. 00:46:23.310 --> 00:46:27.210 align:middle line:84% The liminf always sits below the limsup. 00:46:27.210 --> 00:46:28.620 align:middle line:84% So therefore, those three numbers 00:46:28.620 --> 00:46:31.930 align:middle line:90% have to equal each other. 00:46:31.930 --> 00:46:36.210 align:middle line:84% So this box always sits below this box. 00:46:36.210 --> 00:46:38.940 align:middle line:84% So all three numbers must equal each other then-- 00:46:38.940 --> 00:46:47.180 align:middle line:90% 00:46:47.180 --> 00:46:56.660 align:middle line:84% equals the integral of f equals liminf as n 00:46:56.660 --> 00:47:00.480 align:middle line:90% goes to infinity of fn. 00:47:00.480 --> 00:47:02.940 align:middle line:84% And I bet in analysis, you thought 00:47:02.940 --> 00:47:07.230 align:middle line:84% limsups and liminfs would never be useful, but they are. 00:47:07.230 --> 00:47:12.730 align:middle line:90% 00:47:12.730 --> 00:47:17.740 align:middle line:84% So that is the proof of the dominated convergence theorem. 00:47:17.740 --> 00:47:23.635 align:middle line:84% Now let's use some of this muscle we've been building up. 00:47:23.635 --> 00:47:27.150 align:middle line:90% 00:47:27.150 --> 00:47:33.740 align:middle line:84% So suppose a is less than b and f is a continuous function 00:47:33.740 --> 00:47:34.445 align:middle line:90% on a, b. 00:47:34.445 --> 00:47:40.850 align:middle line:90% 00:47:40.850 --> 00:47:45.675 align:middle line:84% Then the Lesbesgue integral over a, 00:47:45.675 --> 00:47:54.095 align:middle line:84% b of f is equal to the Riemann integral of f. 00:47:54.095 --> 00:47:56.670 align:middle line:90% 00:47:56.670 --> 00:48:01.420 align:middle line:90% This is the Riemann integral. 00:48:01.420 --> 00:48:04.470 align:middle line:90% 00:48:04.470 --> 00:48:06.470 align:middle line:84% So in the course of this proof, we'll 00:48:06.470 --> 00:48:12.335 align:middle line:84% also see why f is, in fact, integrable. 00:48:12.335 --> 00:48:20.100 align:middle line:90% 00:48:20.100 --> 00:48:20.615 align:middle line:90% So, proof. 00:48:20.615 --> 00:48:28.170 align:middle line:90% 00:48:28.170 --> 00:48:32.240 align:middle line:84% So we first show f is Lesbesgue integrable. 00:48:32.240 --> 00:48:42.650 align:middle line:84% So this implies that the absolute value of f 00:48:42.650 --> 00:48:45.620 align:middle line:90% is also a continuous function. 00:48:45.620 --> 00:48:49.370 align:middle line:84% And every continuous function on a closed and bounded interval 00:48:49.370 --> 00:48:50.400 align:middle line:90% is bounded. 00:48:50.400 --> 00:49:00.590 align:middle line:84% So there exists a constant b such that f is-- 00:49:00.590 --> 00:49:02.210 align:middle line:84% the absolute value of f is less than 00:49:02.210 --> 00:49:06.385 align:middle line:84% or equal to b on this closed and bounded interval. 00:49:06.385 --> 00:49:11.420 align:middle line:90% 00:49:11.420 --> 00:49:16.610 align:middle line:84% Then the integral of the absolute value 00:49:16.610 --> 00:49:19.910 align:middle line:84% of f, the Lesbesgue integral of f of the absolute value of f 00:49:19.910 --> 00:49:26.540 align:middle line:84% over a, b, this is less than or equal to a times b, 00:49:26.540 --> 00:49:29.750 align:middle line:84% the integral over a, b of capital P. 00:49:29.750 --> 00:49:32.360 align:middle line:84% And this is the simplest of simple functions. 00:49:32.360 --> 00:49:34.370 align:middle line:84% The Lesbesgue integral of this is just 00:49:34.370 --> 00:49:45.370 align:middle line:84% b times the measure of b of a, b, equals b times b minus a, 00:49:45.370 --> 00:49:47.310 align:middle line:90% which is finite. 00:49:47.310 --> 00:49:50.730 align:middle line:84% So continuous functions are Lesbesgue integrable 00:49:50.730 --> 00:49:52.560 align:middle line:84% on a closed and bounded interval. 00:49:52.560 --> 00:50:01.340 align:middle line:90% 00:50:01.340 --> 00:50:06.170 align:middle line:90% Thus, f is Lesbesgue integrable. 00:50:06.170 --> 00:50:18.240 align:middle line:90% 00:50:18.240 --> 00:50:27.450 align:middle line:84% Now the positive part of f is a continuous function. 00:50:27.450 --> 00:50:31.400 align:middle line:84% And the negative part of f is a continuous function. 00:50:31.400 --> 00:50:33.460 align:middle line:84% So in fact, you can write these down 00:50:33.460 --> 00:50:35.890 align:middle line:84% a little bit differently than I wrote them down before. 00:50:35.890 --> 00:50:37.240 align:middle line:90% f plus is equal to-- 00:50:37.240 --> 00:50:46.030 align:middle line:90% 00:50:46.030 --> 00:50:48.957 align:middle line:90% let's see. 00:50:48.957 --> 00:50:50.790 align:middle line:84% So these are the positive and negative parts 00:50:50.790 --> 00:50:52.860 align:middle line:90% written slightly differently. 00:50:52.860 --> 00:50:55.350 align:middle line:84% If f is continuous, both of these functions 00:50:55.350 --> 00:50:59.460 align:middle line:84% are continuous non-negative functions. 00:50:59.460 --> 00:51:04.170 align:middle line:84% And the Riemann integral of f plus minus f minus, which 00:51:04.170 --> 00:51:07.200 align:middle line:84% is f, the integral of f, is equal to the integral of f 00:51:07.200 --> 00:51:09.600 align:middle line:84% plus minus the integral of f minus, 00:51:09.600 --> 00:51:14.160 align:middle line:84% which is exactly how also Lesbesgue integrals are defined 00:51:14.160 --> 00:51:16.710 align:middle line:84% in terms of the integral, Lesbesgue integral of f 00:51:16.710 --> 00:51:18.420 align:middle line:84% is equal to the Lesbesgue integral 00:51:18.420 --> 00:51:23.000 align:middle line:84% of f plus minus Lesbesgue integral of f minus. 00:51:23.000 --> 00:51:34.699 align:middle line:84% If I consider simply these two cases separately, 00:51:34.699 --> 00:51:41.360 align:middle line:84% be considering these separately and showing 00:51:41.360 --> 00:51:47.060 align:middle line:84% the integral over a, b, f plus or minus 00:51:47.060 --> 00:51:53.130 align:middle line:84% equals the corresponding Riemann integral. 00:51:53.130 --> 00:51:58.770 align:middle line:90% 00:51:58.770 --> 00:52:16.500 align:middle line:84% And using linearity, I may assume that f is non-negative. 00:52:16.500 --> 00:52:18.050 align:middle line:90% So what's the point here? 00:52:18.050 --> 00:52:22.520 align:middle line:84% I'm trying to show this for general continuous functions. 00:52:22.520 --> 00:52:25.880 align:middle line:84% But by splitting it into its positive and negative parts, 00:52:25.880 --> 00:52:29.690 align:middle line:84% it suffices to prove this equals that 00:52:29.690 --> 00:52:31.910 align:middle line:84% for the positive and negative parts, both of which 00:52:31.910 --> 00:52:34.350 align:middle line:84% are continuous functions and non-negative. 00:52:34.350 --> 00:52:38.300 align:middle line:84% So I only need to prove what I want for the case 00:52:38.300 --> 00:52:40.130 align:middle line:90% that f is non-negative. 00:52:40.130 --> 00:52:42.610 align:middle line:90% That's the point. 00:52:42.610 --> 00:52:48.240 align:middle line:84% All right, so we now have a non-negative continuous 00:52:48.240 --> 00:52:49.230 align:middle line:90% function on a, b. 00:52:49.230 --> 00:52:51.540 align:middle line:84% And we want to show the Lesbesgue integral is 00:52:51.540 --> 00:52:55.550 align:middle line:84% equal to the Riemann integral of that continuous function. 00:52:55.550 --> 00:53:08.840 align:middle line:90% So let xn be-- 00:53:08.840 --> 00:53:09.455 align:middle line:90% let's see. 00:53:09.455 --> 00:53:12.170 align:middle line:90% 00:53:12.170 --> 00:53:14.770 align:middle line:90% So this should equal a. 00:53:14.770 --> 00:53:16.840 align:middle line:90% This should equal b-- 00:53:16.840 --> 00:53:30.510 align:middle line:84% be a sequence of partitions of a, 00:53:30.510 --> 00:53:40.970 align:middle line:84% b such that the norm of-- so this is just notation 00:53:40.970 --> 00:53:43.850 align:middle line:90% from back in real analysis. 00:53:43.850 --> 00:53:48.020 align:middle line:84% You shouldn't take this as an actual norm. 00:53:48.020 --> 00:53:50.630 align:middle line:84% Well, I guess it is a norm in a certain sense. 00:53:50.630 --> 00:53:56.600 align:middle line:84% But this is just a subset of a, b that partitions a, b such 00:53:56.600 --> 00:54:00.920 align:middle line:84% that this quantity here, which I denote using the norm-- 00:54:00.920 --> 00:54:05.010 align:middle line:84% but don't confuse us with norms that we discussed before, 00:54:05.010 --> 00:54:07.325 align:middle line:84% which is defined to be the max of-- 00:54:07.325 --> 00:54:11.360 align:middle line:90% 00:54:11.360 --> 00:54:13.580 align:middle line:90% and m could change with n. 00:54:13.580 --> 00:54:20.010 align:middle line:90% 00:54:20.010 --> 00:54:21.930 align:middle line:90% Goes to 0. 00:54:21.930 --> 00:54:29.160 align:middle line:84% So why am I taking a sequence of partitions of a, b? 00:54:29.160 --> 00:54:31.830 align:middle line:84% Because this is how you compute the Riemann integral 00:54:31.830 --> 00:54:33.840 align:middle line:90% in terms of Riemann sums. 00:54:33.840 --> 00:54:36.758 align:middle line:84% And I'm going to show that the sequence of Riemann sums 00:54:36.758 --> 00:54:38.550 align:middle line:84% converging to the Riemann integral actually 00:54:38.550 --> 00:54:40.133 align:middle line:84% converges to the Lesbesgue integral as 00:54:40.133 --> 00:54:45.540 align:middle line:84% well, but along a certain sequence of Riemann sums. 00:54:45.540 --> 00:54:48.760 align:middle line:90% 00:54:48.760 --> 00:54:58.850 align:middle line:84% So let xi j, n in one of the subpartitions. 00:54:58.850 --> 00:55:02.020 align:middle line:90% 00:55:02.020 --> 00:55:05.320 align:middle line:84% So f is a non-negative continuous function 00:55:05.320 --> 00:55:08.870 align:middle line:90% on this interval. 00:55:08.870 --> 00:55:11.710 align:middle line:84% So it has a minimum that it achieves at some point. 00:55:11.710 --> 00:55:23.920 align:middle line:90% 00:55:23.920 --> 00:55:28.360 align:middle line:90% Equals f of xi j, n. 00:55:28.360 --> 00:55:33.340 align:middle line:84% So on the subinterval, f of x is always bigger 00:55:33.340 --> 00:55:35.560 align:middle line:90% than or equal to f of xi n, j. 00:55:35.560 --> 00:55:38.900 align:middle line:90% 00:55:38.900 --> 00:55:43.210 align:middle line:84% So this is how I'm defining the xi n. 00:55:43.210 --> 00:55:47.920 align:middle line:90% 00:55:47.920 --> 00:56:03.020 align:middle line:84% Then by the theory of Riemann integration, 00:56:03.020 --> 00:56:08.390 align:middle line:84% if I look at limit as n goes to infinity 00:56:08.390 --> 00:56:12.215 align:middle line:84% of the associated Riemann sums, this limit exists. 00:56:12.215 --> 00:56:21.200 align:middle line:90% 00:56:21.200 --> 00:56:30.520 align:middle line:84% And you get the Riemann integral of f for a continuous function. 00:56:30.520 --> 00:56:35.050 align:middle line:84% This should have been covered in your introductory analysis 00:56:35.050 --> 00:56:36.145 align:middle line:90% class that-- 00:56:36.145 --> 00:56:38.860 align:middle line:90% 00:56:38.860 --> 00:56:41.950 align:middle line:84% if you like, this is the lower Riemann integral 00:56:41.950 --> 00:56:43.540 align:middle line:90% or lower Riemann sum. 00:56:43.540 --> 00:56:47.470 align:middle line:84% And that as long as you're going along a partition 00:56:47.470 --> 00:56:50.650 align:middle line:84% so that this quantity here is going to 0, 00:56:50.650 --> 00:56:53.595 align:middle line:84% then the associated Riemann sums converge 00:56:53.595 --> 00:56:54.595 align:middle line:90% to the Riemann integral. 00:56:54.595 --> 00:56:59.330 align:middle line:90% 00:56:59.330 --> 00:57:04.795 align:middle line:84% All right, now each of these is a finite set. 00:57:04.795 --> 00:57:10.130 align:middle line:90% 00:57:10.130 --> 00:57:14.800 align:middle line:84% Let n be the union of these sets. 00:57:14.800 --> 00:57:21.570 align:middle line:90% 00:57:21.570 --> 00:57:24.570 align:middle line:84% Then this is a countable union of finite sets. 00:57:24.570 --> 00:57:26.745 align:middle line:90% So it's countable. 00:57:26.745 --> 00:57:31.340 align:middle line:84% A countable union of countable sets is countable. 00:57:31.340 --> 00:57:36.860 align:middle line:84% In particular, this means the measure of this set is 0. 00:57:36.860 --> 00:57:40.910 align:middle line:84% So if I-- the set of all partition points 00:57:40.910 --> 00:57:47.270 align:middle line:84% as I range over all of the partitions converging to 0-- so 00:57:47.270 --> 00:57:49.070 align:middle line:84% I just took any sequence of partitions 00:57:49.070 --> 00:57:51.497 align:middle line:84% with this quantity here going to 0. 00:57:51.497 --> 00:57:53.330 align:middle line:84% If I take the union of all these partitions, 00:57:53.330 --> 00:57:54.440 align:middle line:90% I get a countable set. 00:57:54.440 --> 00:57:56.780 align:middle line:84% That set has measure 0 because it's countable. 00:57:56.780 --> 00:58:06.120 align:middle line:90% 00:58:06.120 --> 00:58:08.820 align:middle line:84% Why am I making this point that it has measure 0? 00:58:08.820 --> 00:58:12.795 align:middle line:84% Well, because off of this set, magic happens. 00:58:12.795 --> 00:58:21.210 align:middle line:90% 00:58:21.210 --> 00:58:26.490 align:middle line:84% And what we've learned is that magic happening off 00:58:26.490 --> 00:58:32.190 align:middle line:84% of a set of measure 0 means magic happens for integrals. 00:58:32.190 --> 00:58:45.110 align:middle line:90% 00:58:45.110 --> 00:58:47.360 align:middle line:90% So let me-- 00:58:47.360 --> 00:58:49.400 align:middle line:84% I'd like one more important piece of information 00:58:49.400 --> 00:58:52.640 align:middle line:84% that we have from the theory of Riemann 00:58:52.640 --> 00:58:57.155 align:middle line:84% integration that the Riemann sums converge to a Riemann 00:58:57.155 --> 00:58:57.655 align:middle line:90% integral. 00:58:57.655 --> 00:59:05.470 align:middle line:90% 00:59:05.470 --> 00:59:08.530 align:middle line:84% Let fn be the following simple function. 00:59:08.530 --> 00:59:13.630 align:middle line:84% This is sum from j equals 1 to mn 00:59:13.630 --> 00:59:21.200 align:middle line:84% of f of c in j of the indicator-- times 00:59:21.200 --> 00:59:35.220 align:middle line:84% the indicator function of xjn j minus 1 xjn. 00:59:35.220 --> 00:59:43.825 align:middle line:84% And then could put plus 0 times the indicator function of xjn. 00:59:43.825 --> 00:59:47.100 align:middle line:90% 00:59:47.100 --> 00:59:48.833 align:middle line:84% I mean, this part doesn't really matter. 00:59:48.833 --> 00:59:49.500 align:middle line:90% I'm just saying. 00:59:49.500 --> 00:59:53.490 align:middle line:84% So this is a simple function for each n. 00:59:53.490 --> 01:00:02.455 align:middle line:90% 01:00:02.455 --> 01:00:04.205 align:middle line:84% Should say a non-negative simple function. 01:00:04.205 --> 01:00:13.650 align:middle line:90% 01:00:13.650 --> 01:00:18.480 align:middle line:90% So what's happening now? 01:00:18.480 --> 01:00:20.400 align:middle line:90% And why did I choose the xi n? 01:00:20.400 --> 01:00:22.515 align:middle line:84% So let me draw the picture that goes with this. 01:00:22.515 --> 01:00:26.450 align:middle line:90% 01:00:26.450 --> 01:00:31.310 align:middle line:90% I have my function f on a, b. 01:00:31.310 --> 01:00:35.510 align:middle line:84% And what I'm doing is I'm cutting up 01:00:35.510 --> 01:00:43.525 align:middle line:84% the domain to get the approximate Riemann integral. 01:00:43.525 --> 01:01:00.157 align:middle line:90% 01:01:00.157 --> 01:01:01.115 align:middle line:90% So that should connect. 01:01:01.115 --> 01:01:04.900 align:middle line:90% 01:01:04.900 --> 01:01:07.990 align:middle line:84% And I'm choosing the heights to be the minimum of f 01:01:07.990 --> 01:01:10.480 align:middle line:90% on each of these intervals. 01:01:10.480 --> 01:01:19.060 align:middle line:84% So this is a f of xi 1, if you like, f of xi 2. 01:01:19.060 --> 01:01:21.850 align:middle line:84% And at least for this picture, this is xi 1. 01:01:21.850 --> 01:01:26.460 align:middle line:90% 01:01:26.460 --> 01:01:30.950 align:middle line:84% And what I know is as I'm making this partition finer and finer, 01:01:30.950 --> 01:01:38.020 align:middle line:84% these approximate areas here are converging to the full Riemann 01:01:38.020 --> 01:01:40.730 align:middle line:90% integral of f. 01:01:40.730 --> 01:01:43.020 align:middle line:84% Another way to think about that is 01:01:43.020 --> 01:01:48.880 align:middle line:84% that what these are just the Lesbesgue 01:01:48.880 --> 01:01:51.250 align:middle line:84% integral of certain simple functions, where 01:01:51.250 --> 01:01:55.060 align:middle line:84% the simple function is 1 on this and has height f of xi 01:01:55.060 --> 01:01:57.650 align:middle line:90% in or xi 1. 01:01:57.650 --> 01:02:00.250 align:middle line:84% This quantity here, the integral-- the area 01:02:00.250 --> 01:02:03.280 align:middle line:84% underneath this-- is equal to the Lesbesgue integral 01:02:03.280 --> 01:02:09.110 align:middle line:84% of f of xi 2 times the indicator function of this interval, 01:02:09.110 --> 01:02:10.160 align:middle line:90% and so on. 01:02:10.160 --> 01:02:16.670 align:middle line:84% So I can view these pieces here that are entering 01:02:16.670 --> 01:02:20.030 align:middle line:84% in the Riemann sum as Lesbesgue integrals 01:02:20.030 --> 01:02:21.350 align:middle line:90% of certain simple functions. 01:02:21.350 --> 01:02:23.930 align:middle line:84% Or I can view this entire quantity 01:02:23.930 --> 01:02:29.330 align:middle line:84% as the Lesbesgue integral of a simple function for reach n. 01:02:29.330 --> 01:02:31.880 align:middle line:84% Now the goal here is what we'll do 01:02:31.880 --> 01:02:35.810 align:middle line:84% is we'll show that the simple functions whose-- 01:02:35.810 --> 01:02:39.080 align:middle line:84% I mean, in fact, we can do just do this now. 01:02:39.080 --> 01:02:48.850 align:middle line:84% Note, for all n, if I look at the Lesbesgue integral of fn 01:02:48.850 --> 01:02:54.940 align:middle line:84% over a, b, this is equal to sum from j equals 1 to m sub n. 01:02:54.940 --> 01:02:57.940 align:middle line:84% This is a set of measure 0, so it doesn't contribute. 01:02:57.940 --> 01:02:59.703 align:middle line:90% f of xi jn-- 01:02:59.703 --> 01:03:01.120 align:middle line:84% these are all non-negative numbers 01:03:01.120 --> 01:03:06.650 align:middle line:84% because f is non-negative-- times the measure of xj 01:03:06.650 --> 01:03:19.920 align:middle line:90% minus 1n, xjn times xjn. 01:03:19.920 --> 01:03:23.800 align:middle line:84% Now remember, we built up Lesbesgue integration 01:03:23.800 --> 01:03:25.400 align:middle line:84% so that the measure of an interval 01:03:25.400 --> 01:03:26.650 align:middle line:90% is the length of the interval. 01:03:26.650 --> 01:03:31.320 align:middle line:84% So this is equal to xjn minus xj minus 1n. 01:03:31.320 --> 01:03:36.900 align:middle line:84% So the integral, the Lesbesgue integral, 01:03:36.900 --> 01:03:44.810 align:middle line:84% of each of these functions, each of these simple functions, 01:03:44.810 --> 01:03:51.920 align:middle line:84% equals that Riemann sum appearing in this limit, right? 01:03:51.920 --> 01:03:57.020 align:middle line:84% And now the goal is to show that the fns converge 01:03:57.020 --> 01:03:59.090 align:middle line:84% to f almost everywhere and are bounded 01:03:59.090 --> 01:04:02.070 align:middle line:84% above by an integral function at least almost everywhere. 01:04:02.070 --> 01:04:04.670 align:middle line:84% Then I can apply the dominated convergence theorem 01:04:04.670 --> 01:04:07.010 align:middle line:84% to conclude that the limit n m goes 01:04:07.010 --> 01:04:11.840 align:middle line:84% to infinity of this thing equals the Lesbesgue integral of f. 01:04:11.840 --> 01:04:14.780 align:middle line:84% But this, the limit as n goes to infinity of this thing, 01:04:14.780 --> 01:04:16.850 align:middle line:84% is equal to the limit as n goes to infinity 01:04:16.850 --> 01:04:20.840 align:middle line:84% of this quantity, which is equal to the Riemann integral. 01:04:20.840 --> 01:04:22.385 align:middle line:90% And that's the game plan. 01:04:22.385 --> 01:04:27.880 align:middle line:90% 01:04:27.880 --> 01:04:29.750 align:middle line:84% And I mean, you can already guess from here. 01:04:29.750 --> 01:04:32.050 align:middle line:84% What's the function that sits above all of the fns? 01:04:32.050 --> 01:04:35.950 align:middle line:90% 01:04:35.950 --> 01:04:40.426 align:middle line:84% It's going to be at least away from these-- 01:04:40.426 --> 01:04:44.160 align:middle line:84% at least away from possibly the endpoints. 01:04:44.160 --> 01:04:44.955 align:middle line:90% It's going to be f. 01:04:44.955 --> 01:04:47.910 align:middle line:90% 01:04:47.910 --> 01:04:54.870 align:middle line:84% Then for all xn a, b, take away n. 01:04:54.870 --> 01:04:58.350 align:middle line:90% A couple of things I have. 01:04:58.350 --> 01:05:06.785 align:middle line:84% Well, first thing is less than or equal to f of x. 01:05:06.785 --> 01:05:10.980 align:middle line:90% 01:05:10.980 --> 01:05:15.120 align:middle line:84% Now I'm going to show that on a, b, take away 01:05:15.120 --> 01:05:18.600 align:middle line:84% n, which is a set of measure 0, that fn of x 01:05:18.600 --> 01:05:19.800 align:middle line:90% converges to f of x. 01:05:19.800 --> 01:05:22.550 align:middle line:90% 01:05:22.550 --> 01:05:23.240 align:middle line:90% So now I claim-- 01:05:23.240 --> 01:05:32.740 align:middle line:90% 01:05:32.740 --> 01:05:43.220 align:middle line:84% then-- i.e., fn converges to f almost everywhere. 01:05:43.220 --> 01:05:46.420 align:middle line:84% And it's bounded above by a Lesbesgue integral function 01:05:46.420 --> 01:05:47.330 align:middle line:90% almost everywhere. 01:05:47.330 --> 01:05:49.750 align:middle line:84% So then we can apply the dominated convergence theorem 01:05:49.750 --> 01:05:50.980 align:middle line:90% to get what we want. 01:05:50.980 --> 01:05:52.330 align:middle line:90% So let's prove this. 01:05:52.330 --> 01:05:56.890 align:middle line:90% 01:05:56.890 --> 01:06:01.455 align:middle line:84% So let xb and ab take away the set of all partition points. 01:06:01.455 --> 01:06:04.770 align:middle line:90% 01:06:04.770 --> 01:06:06.110 align:middle line:90% So we want to show this. 01:06:06.110 --> 01:06:08.420 align:middle line:84% Let's just go back to a basic epsilon. 01:06:08.420 --> 01:06:10.760 align:middle line:84% In argument, let epsilon be positive. 01:06:10.760 --> 01:06:14.970 align:middle line:90% 01:06:14.970 --> 01:06:24.530 align:middle line:84% Since f is continuous at x, there 01:06:24.530 --> 01:06:33.230 align:middle line:84% exists a delta positive so that if x minus y 01:06:33.230 --> 01:06:42.000 align:middle line:84% is less than delta, then f of x minus f of y 01:06:42.000 --> 01:06:42.875 align:middle line:90% is less than epsilon. 01:06:42.875 --> 01:06:46.240 align:middle line:90% 01:06:46.240 --> 01:06:48.880 align:middle line:84% Now we know that the partitions are getting finer and finer, 01:06:48.880 --> 01:06:49.380 align:middle line:90% right? 01:06:49.380 --> 01:06:54.330 align:middle line:84% So x is in a, b take away n, right? 01:06:54.330 --> 01:07:02.040 align:middle line:84% So since the norms of these partitions, which remember, 01:07:02.040 --> 01:07:02.775 align:middle line:90% is the max-- 01:07:02.775 --> 01:07:14.750 align:middle line:90% 01:07:14.750 --> 01:07:16.930 align:middle line:84% I forgot that n when I was writing it here. 01:07:16.930 --> 01:07:18.520 align:middle line:90% And this-- should be n. 01:07:18.520 --> 01:07:22.740 align:middle line:90% 01:07:22.740 --> 01:07:30.720 align:middle line:84% Since this goes to 0, there exist capital M 01:07:30.720 --> 01:07:35.670 align:middle line:84% so that all n bigger than or equal to capital M, 01:07:35.670 --> 01:07:42.560 align:middle line:84% this quantity here, the length of-- 01:07:42.560 --> 01:07:44.570 align:middle line:84% the longest length of the subintervals-- 01:07:44.570 --> 01:07:48.760 align:middle line:90% 01:07:48.760 --> 01:07:50.660 align:middle line:90% is less than delta. 01:07:50.660 --> 01:07:52.130 align:middle line:84% So now let me draw you a picture. 01:07:52.130 --> 01:07:55.660 align:middle line:84% So x is in a, b, take away the partitions, 01:07:55.660 --> 01:07:59.720 align:middle line:84% all the possible partition points, all the xj's, n's, 01:07:59.720 --> 01:08:00.220 align:middle line:90% right? 01:08:00.220 --> 01:08:03.050 align:middle line:90% 01:08:03.050 --> 01:08:11.680 align:middle line:84% So let-- I claim now that fn of x minus f of x 01:08:11.680 --> 01:08:12.760 align:middle line:90% is less than epsilon. 01:08:12.760 --> 01:08:15.880 align:middle line:90% 01:08:15.880 --> 01:08:26.309 align:middle line:84% Now how do I evaluate for x in a, b take away n fn of x? 01:08:26.309 --> 01:08:29.580 align:middle line:90% 01:08:29.580 --> 01:08:34.475 align:middle line:90% And fn of x is equal to-- 01:08:34.475 --> 01:08:43.220 align:middle line:90% 01:08:43.220 --> 01:08:51.245 align:middle line:84% times the indicator function of xj minus 1n, xjn of x. 01:08:51.245 --> 01:08:53.870 align:middle line:84% And I don't have to worry about the endpoint, because remember, 01:08:53.870 --> 01:08:58.940 align:middle line:84% x is in a, b, take away all of the partition points. 01:08:58.940 --> 01:09:03.819 align:middle line:90% 01:09:03.819 --> 01:09:07.590 align:middle line:84% xjn is always a partition point. xjn is always 01:09:07.590 --> 01:09:11.370 align:middle line:84% b, so I'm always taking away b and a. 01:09:11.370 --> 01:09:12.930 align:middle line:90% So this must equal-- 01:09:12.930 --> 01:09:15.279 align:middle line:84% this x must lie in one of these intervals, 01:09:15.279 --> 01:09:19.050 align:middle line:84% but not be one of the partition points. 01:09:19.050 --> 01:09:21.569 align:middle line:90% So this must equal-- 01:09:21.569 --> 01:09:23.160 align:middle line:90% this is bad notation. 01:09:23.160 --> 01:09:25.709 align:middle line:90% But let's say fxe, OK, k-- 01:09:25.709 --> 01:09:28.560 align:middle line:90% 01:09:28.560 --> 01:09:52.450 align:middle line:84% for the unique k, such that x is in xk minus 1n xk. 01:09:52.450 --> 01:10:01.090 align:middle line:90% 01:10:01.090 --> 01:10:14.630 align:middle line:84% Then since xi n, k is in this interval, 01:10:14.630 --> 01:10:19.430 align:middle line:84% and the max over the length of the small intervals, 01:10:19.430 --> 01:10:23.300 align:middle line:84% which is the differences here, is less than delta-- 01:10:23.300 --> 01:10:24.940 align:middle line:90% so in particular for this one. 01:10:24.940 --> 01:10:27.990 align:middle line:90% 01:10:27.990 --> 01:10:30.750 align:middle line:90% So I have-- here's a picture. 01:10:30.750 --> 01:10:35.610 align:middle line:90% Here's xk minus 1n, xnk. 01:10:35.610 --> 01:10:38.370 align:middle line:90% 01:10:38.370 --> 01:10:41.670 align:middle line:90% Xi nk is somewhere in there. 01:10:41.670 --> 01:10:43.050 align:middle line:90% x is somewhere in there. 01:10:43.050 --> 01:10:46.350 align:middle line:90% 01:10:46.350 --> 01:10:49.440 align:middle line:84% And since this quantity here is less than delta, 01:10:49.440 --> 01:11:01.060 align:middle line:84% this implies that x minus xi in k must be less than delta. 01:11:01.060 --> 01:11:07.840 align:middle line:84% And therefore, f of x minus f xi f 01:11:07.840 --> 01:11:17.350 align:middle line:84% sub n of x, which is equal to f of x minus f of xi kn. 01:11:17.350 --> 01:11:21.790 align:middle line:84% Now these are within delta distance to each other. 01:11:21.790 --> 01:11:25.060 align:middle line:84% And therefore, f of x and f of that number 01:11:25.060 --> 01:11:27.430 align:middle line:84% must be within epsilon of each other 01:11:27.430 --> 01:11:29.305 align:middle line:90% simply by how delta was chosen. 01:11:29.305 --> 01:11:33.540 align:middle line:90% 01:11:33.540 --> 01:11:36.600 align:middle line:84% So we've shown that for all n bigger than or equal to capital 01:11:36.600 --> 01:11:41.190 align:middle line:84% M, f of x minus fn of x is less than epsilon. 01:11:41.190 --> 01:11:43.800 align:middle line:90% 01:11:43.800 --> 01:11:48.330 align:middle line:84% Thus, limit as n goes to infinity 01:11:48.330 --> 01:11:56.850 align:middle line:84% of fn of x equals f of x, all x in a, b, 01:11:56.850 --> 01:11:59.130 align:middle line:90% take away the partition points. 01:11:59.130 --> 01:12:14.250 align:middle line:90% 01:12:14.250 --> 01:12:24.730 align:middle line:90% So we have a couple of things. 01:12:24.730 --> 01:12:27.540 align:middle line:84% So I said this, but now I'm just going to write down what. 01:12:27.540 --> 01:12:30.900 align:middle line:90% 01:12:30.900 --> 01:12:32.130 align:middle line:90% We have these two things. 01:12:32.130 --> 01:12:39.850 align:middle line:84% We have that almost everywhere, fn's are converging to f. 01:12:39.850 --> 01:12:42.460 align:middle line:84% Almost everywhere, fn is less than or equal to f. 01:12:42.460 --> 01:12:45.490 align:middle line:84% f is a fixed continuous function that's integrable. 01:12:45.490 --> 01:12:47.630 align:middle line:90% It's non-negative. 01:12:47.630 --> 01:12:54.750 align:middle line:84% So by the dominated convergence theorem, 01:12:54.750 --> 01:12:59.370 align:middle line:84% the integral, Lesbesgue integral, 01:12:59.370 --> 01:13:03.210 align:middle line:84% of this function, continuous function f of x, 01:13:03.210 --> 01:13:06.450 align:middle line:84% is equal to the limit as n goes to infinity 01:13:06.450 --> 01:13:13.070 align:middle line:84% of the Lesbesgue integrals of these special simple functions, 01:13:13.070 --> 01:13:25.930 align:middle line:84% which, as we computed right here, is equal to the limit 01:13:25.930 --> 01:13:29.680 align:middle line:84% as n goes to infinity of k equals 1 to m 01:13:29.680 --> 01:13:38.500 align:middle line:84% sub n, f of xi in j minus xj and minus xj minus 1n. 01:13:38.500 --> 01:13:47.560 align:middle line:90% And by how this-- 01:13:47.560 --> 01:13:49.410 align:middle line:84% what we know about Riemann integrals, 01:13:49.410 --> 01:13:51.060 align:middle line:84% this converges to the Riemann integral. 01:13:51.060 --> 01:13:58.170 align:middle line:90% 01:13:58.170 --> 01:14:03.372 align:middle line:84% And thus the Lesbesgue integral equals the Riemann integral 01:14:03.372 --> 01:14:04.455 align:middle line:90% for a continuous function. 01:14:04.455 --> 01:14:14.670 align:middle line:90% 01:14:14.670 --> 01:14:19.050 align:middle line:84% So we have discussed the Riemann integral 01:14:19.050 --> 01:14:23.460 align:middle line:84% of real-valued measurable functions 01:14:23.460 --> 01:14:28.570 align:middle line:84% or real-valued integrable functions. 01:14:28.570 --> 01:14:33.220 align:middle line:84% Now quite often, we want to have complex-valued functions 01:14:33.220 --> 01:14:37.150 align:middle line:84% defined on measurable subsets of real numbers. 01:14:37.150 --> 01:14:39.920 align:middle line:90% What do we do then? 01:14:39.920 --> 01:14:41.860 align:middle line:84% Well, everything that we've done basically 01:14:41.860 --> 01:14:44.980 align:middle line:84% applies as long as it makes sense. 01:14:44.980 --> 01:14:47.920 align:middle line:90% So let me state that here. 01:14:47.920 --> 01:14:56.230 align:middle line:90% 01:14:56.230 --> 01:14:58.320 align:middle line:84% So all of the previous theorems that we've 01:14:58.320 --> 01:15:14.900 align:middle line:84% proven, as long as they make sense, 01:15:14.900 --> 01:15:18.380 align:middle line:84% you can use what we've done for real-valued integrable 01:15:18.380 --> 01:15:21.530 align:middle line:84% functions to what we call complex-valued integrable 01:15:21.530 --> 01:15:37.430 align:middle line:84% function imply the corresponding statements 01:15:37.430 --> 01:15:51.740 align:middle line:84% for complex-valued integrable functions. 01:15:51.740 --> 01:15:52.835 align:middle line:90% And what are these? 01:15:52.835 --> 01:15:56.830 align:middle line:90% 01:15:56.830 --> 01:16:02.800 align:middle line:84% So f from a measurable subset of E to now the complex numbers. 01:16:02.800 --> 01:16:06.625 align:middle line:90% 01:16:06.625 --> 01:16:13.360 align:middle line:90% So let me say E to-- 01:16:13.360 --> 01:16:17.230 align:middle line:84% and now let me just define what I mean by complex integrable 01:16:17.230 --> 01:16:20.115 align:middle line:90% functions, meaning-- 01:16:20.115 --> 01:16:24.300 align:middle line:90% 01:16:24.300 --> 01:16:35.840 align:middle line:84% so f from a measurable subset of E to C is Lesbesgue integrable 01:16:35.840 --> 01:16:44.670 align:middle line:84% if the same condition holds, that the integral 01:16:44.670 --> 01:16:48.270 align:middle line:90% of the absolute value of f is-- 01:16:48.270 --> 01:16:50.730 align:middle line:84% or now this is a modulus of a complex number 01:16:50.730 --> 01:16:52.930 align:middle line:90% for each x-- is integrable. 01:16:52.930 --> 01:16:57.760 align:middle line:84% So this is a non-negative measurable function now 01:16:57.760 --> 01:17:01.210 align:middle line:84% which is defined on E. So this quantity here makes sense. 01:17:01.210 --> 01:17:03.690 align:middle line:84% So we say a complex-valued function 01:17:03.690 --> 01:17:05.985 align:middle line:84% is Lesbesgue integrable if this quantity is finite. 01:17:05.985 --> 01:17:08.570 align:middle line:90% 01:17:08.570 --> 01:17:20.140 align:middle line:84% And now the definition of the Lesbesgue integral 01:17:20.140 --> 01:17:23.590 align:middle line:84% of a complex-valued integrable function 01:17:23.590 --> 01:17:31.340 align:middle line:84% is defined to be simply the integral of the real part of f, 01:17:31.340 --> 01:17:37.370 align:middle line:84% which is, again, a real-valued integrable function, 01:17:37.370 --> 01:17:42.920 align:middle line:84% plus i times the integral of the imaginary part of f. 01:17:42.920 --> 01:17:49.460 align:middle line:84% So again, we just-- to define complex integrable functions 01:17:49.460 --> 01:17:51.890 align:middle line:84% or complex-valued integrable functions, 01:17:51.890 --> 01:17:53.810 align:middle line:84% we just take the same definition. 01:17:53.810 --> 01:17:56.150 align:middle line:84% And then we define the integral to be 01:17:56.150 --> 01:18:00.440 align:middle line:84% the integral of the real part plus i times the integral 01:18:00.440 --> 01:18:01.560 align:middle line:90% of the imaginary part. 01:18:01.560 --> 01:18:03.740 align:middle line:84% And the fact that this is finite implies 01:18:03.740 --> 01:18:06.890 align:middle line:84% that these two real-valued measurable functions 01:18:06.890 --> 01:18:08.644 align:middle line:90% are, in fact, integrable. 01:18:08.644 --> 01:18:13.290 align:middle line:90% 01:18:13.290 --> 01:18:17.790 align:middle line:84% And all of the theorems that we stated before, as long as they 01:18:17.790 --> 01:18:20.445 align:middle line:90% make sense, carry over. 01:18:20.445 --> 01:18:25.020 align:middle line:84% For example, linearity of the integral with respect 01:18:25.020 --> 01:18:28.110 align:middle line:84% to now multiplication by complex numbers carries over. 01:18:28.110 --> 01:18:30.420 align:middle line:84% The integral of the sum is the sum of the integrals. 01:18:30.420 --> 01:18:37.600 align:middle line:84% That still carries over just by using this definition, 01:18:37.600 --> 01:18:41.680 align:middle line:84% along with the fact that we know that the integral of two-- 01:18:41.680 --> 01:18:44.470 align:middle line:84% the sum of two real-valued functions is integrable. 01:18:44.470 --> 01:18:48.730 align:middle line:84% And the Lesbesgue dominated convergence theorem then 01:18:48.730 --> 01:18:49.480 align:middle line:90% carries-- 01:18:49.480 --> 01:18:53.620 align:middle line:84% can be generalized to complex-valued integrable 01:18:53.620 --> 01:18:55.210 align:middle line:90% functions. 01:18:55.210 --> 01:18:59.380 align:middle line:84% Maybe what's not so clear is the-- 01:18:59.380 --> 01:19:01.420 align:middle line:84% or let me just at least give you the flavor 01:19:01.420 --> 01:19:04.360 align:middle line:84% of how you can use what you know about real-valued integrable 01:19:04.360 --> 01:19:07.897 align:middle line:84% functions to get corresponding statements 01:19:07.897 --> 01:19:09.480 align:middle line:84% for complex-valued integral functions. 01:19:09.480 --> 01:19:14.330 align:middle line:84% So let's do the triangle inequality for integrals, 01:19:14.330 --> 01:19:14.830 align:middle line:90% right? 01:19:14.830 --> 01:19:22.550 align:middle line:84% So if I have a complex integrable or complex-valued 01:19:22.550 --> 01:19:24.160 align:middle line:90% integral-- 01:19:24.160 --> 01:19:32.820 align:middle line:84% complex-valued integrable function-- then the integral 01:19:32.820 --> 01:19:35.880 align:middle line:84% of E of f-- so this is now a complex number. 01:19:35.880 --> 01:19:41.890 align:middle line:84% Taking the modulus of that is less than or equal to that. 01:19:41.890 --> 01:19:47.940 align:middle line:90% So what's the proof? 01:19:47.940 --> 01:19:49.660 align:middle line:90% So this is clear. 01:19:49.660 --> 01:19:58.430 align:middle line:84% So if the integral of E equals 0. 01:19:58.430 --> 01:20:00.120 align:middle line:84% So that's just clear automatically. 01:20:00.120 --> 01:20:01.625 align:middle line:84% So let's assume it's not equal to 0. 01:20:01.625 --> 01:20:09.960 align:middle line:90% 01:20:09.960 --> 01:20:15.720 align:middle line:84% And let alpha be the following complex number, integral 01:20:15.720 --> 01:20:18.860 align:middle line:90% of E over-- 01:20:18.860 --> 01:20:22.390 align:middle line:90% 01:20:22.390 --> 01:20:26.590 align:middle line:84% so the integral of f over E is a complex number. 01:20:26.590 --> 01:20:27.970 align:middle line:90% I take its complex conjugate. 01:20:27.970 --> 01:20:29.770 align:middle line:90% I divide by the modulus of that. 01:20:29.770 --> 01:20:35.890 align:middle line:90% 01:20:35.890 --> 01:20:41.020 align:middle line:90% Then alpha and modulus equals 1. 01:20:41.020 --> 01:20:43.650 align:middle line:90% 01:20:43.650 --> 01:20:51.080 align:middle line:84% And the absolute value of the integral of f-- this 01:20:51.080 --> 01:20:59.570 align:middle line:84% is equal to alpha times the integral of f over E 01:20:59.570 --> 01:21:02.210 align:middle line:84% because the complex conjugate times this 01:21:02.210 --> 01:21:04.820 align:middle line:84% gives me the modulus squared divided by the modulus. 01:21:04.820 --> 01:21:07.490 align:middle line:90% I get back the modulus, right? 01:21:07.490 --> 01:21:11.300 align:middle line:84% And linearity of the Riemann-- or not Riemann-- 01:21:11.300 --> 01:21:14.420 align:middle line:84% of the Lesbesguen rule for complex-valued integrable 01:21:14.420 --> 01:21:15.450 align:middle line:90% functions still holds. 01:21:15.450 --> 01:21:20.630 align:middle line:90% So I can pull that alpha inside. 01:21:20.630 --> 01:21:24.150 align:middle line:84% And now this thing is equal to this. 01:21:24.150 --> 01:21:25.590 align:middle line:90% So it's equal to a real number. 01:21:25.590 --> 01:21:26.915 align:middle line:90% So it's equal to its real part. 01:21:26.915 --> 01:21:33.630 align:middle line:90% 01:21:33.630 --> 01:21:36.480 align:middle line:84% And now the real part of the integral 01:21:36.480 --> 01:21:38.310 align:middle line:84% is equal to the integral of the real part 01:21:38.310 --> 01:21:40.380 align:middle line:90% just by the definition. 01:21:40.380 --> 01:21:46.546 align:middle line:84% So this is equal to the real part of alpha times f. 01:21:46.546 --> 01:21:50.550 align:middle line:90% 01:21:50.550 --> 01:21:56.800 align:middle line:84% Now this is a real-valued function, integrable function. 01:21:56.800 --> 01:22:01.140 align:middle line:84% And we know from what we proved for real-valued integrable 01:22:01.140 --> 01:22:06.000 align:middle line:84% functions, that's less than or equal to the real part 01:22:06.000 --> 01:22:14.440 align:middle line:84% of alpha times f over E. Now the absolute value of the real part 01:22:14.440 --> 01:22:17.680 align:middle line:84% of a complex number is less than or equal to the modulus 01:22:17.680 --> 01:22:18.775 align:middle line:90% of that complex number. 01:22:18.775 --> 01:22:25.880 align:middle line:90% 01:22:25.880 --> 01:22:29.480 align:middle line:84% And alpha has modulus equal to 1. 01:22:29.480 --> 01:22:39.120 align:middle line:84% So this is equal to the integral of the absolute value. 01:22:39.120 --> 01:22:42.320 align:middle line:84% So at this point right here, we used 01:22:42.320 --> 01:22:45.500 align:middle line:84% what we knew about real-valued integrable functions. 01:22:45.500 --> 01:22:49.190 align:middle line:84% And using that, we then got the corresponding statement 01:22:49.190 --> 01:22:52.940 align:middle line:84% for complex-valued integrable functions. 01:22:52.940 --> 01:22:57.440 align:middle line:84% And using the theorems that we proved 01:22:57.440 --> 01:22:59.300 align:middle line:84% for real-valued integrable functions, 01:22:59.300 --> 01:23:02.780 align:middle line:84% we get corresponding statements for complex-valued integrable 01:23:02.780 --> 01:23:05.270 align:middle line:84% functions as long as the statements make sense. 01:23:05.270 --> 01:23:09.015 align:middle line:84% We don't say two complex-valued functions, 01:23:09.015 --> 01:23:10.640 align:middle line:84% one is less than or equal to the other, 01:23:10.640 --> 01:23:12.630 align:middle line:84% because we don't have an order on complex numbers. 01:23:12.630 --> 01:23:14.172 align:middle line:84% So that statement doesn't make sense. 01:23:14.172 --> 01:23:23.380 align:middle line:84% But statements about, for example, 01:23:23.380 --> 01:23:25.740 align:middle line:84% complex-valued continuous functions, the Lesbesgue 01:23:25.740 --> 01:23:28.737 align:middle line:84% integral equaling the Riemann integral, that 01:23:28.737 --> 01:23:30.570 align:middle line:84% follows just immediately because the Riemann 01:23:30.570 --> 01:23:33.360 align:middle line:84% integral of a complex number-- a complex-valued continuous 01:23:33.360 --> 01:23:36.400 align:middle line:84% function-- is just defined to be the same thing. 01:23:36.400 --> 01:23:37.650 align:middle line:90% And since we know about-- 01:23:37.650 --> 01:23:42.420 align:middle line:84% know that this equals the Riemann 01:23:42.420 --> 01:23:45.930 align:middle line:84% integral of the real part of f, and this equals the Riemann 01:23:45.930 --> 01:23:47.790 align:middle line:84% integral of the imaginary part of f, which 01:23:47.790 --> 01:23:52.320 align:middle line:84% is by definition equal to the Riemann integral of f, 01:23:52.320 --> 01:23:55.770 align:middle line:84% then we immediately get the previous theorem generalized 01:23:55.770 --> 01:24:02.150 align:middle line:90% to complex-valued functions. 01:24:02.150 --> 01:24:06.402 align:middle line:84% All right, so that's the theory of Riemann integration. 01:24:06.402 --> 01:24:07.610 align:middle line:90% I mean-- Riemann integration? 01:24:07.610 --> 01:24:09.330 align:middle line:90% Lesbesgue integration. 01:24:09.330 --> 01:24:15.830 align:middle line:84% Next time, we will finish our discussion of-- 01:24:15.830 --> 01:24:19.590 align:middle line:90% 01:24:19.590 --> 01:24:22.050 align:middle line:84% measure an integration by introducing the big Lp 01:24:22.050 --> 01:24:26.610 align:middle line:84% spaces, which are spaces of measurable functions 01:24:26.610 --> 01:24:27.675 align:middle line:90% which have a finite-- 01:24:27.675 --> 01:24:30.960 align:middle line:90% 01:24:30.960 --> 01:24:34.290 align:middle line:84% which raised to a certain power, have a finite Lesbesguen rule, 01:24:34.290 --> 01:24:40.680 align:middle line:84% and show that those are Banach spaces that 01:24:40.680 --> 01:24:43.880 align:middle line:84% contain the continuous functions, of course, 01:24:43.880 --> 01:24:46.700 align:middle line:90% and in certain cases are-- 01:24:46.700 --> 01:24:48.800 align:middle line:84% the space of continuous functions 01:24:48.800 --> 01:24:51.070 align:middle line:90% are dense in these spaces. 01:24:51.070 --> 01:24:57.000 align:middle line:90%