
Title: Measurable Maps

Series: Measure Theory

YouTubeTitle: Measure Theory 5  Measurable Maps

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Subtitle in English
1 00:00:00,589 –> 00:00:02,539 Hello and welcome back to
2 00:00:02,549 –> 00:00:03,769 measure theory.
3 00:00:04,070 –> 00:00:06,000 We have reached part 5
4 00:00:06,010 –> 00:00:07,119 in our series.
5 00:00:07,130 –> 00:00:08,609 And today I want to talk
6 00:00:08,619 –> 00:00:10,220 about measurable
7 00:00:10,229 –> 00:00:10,880 maps.
8 00:00:11,699 –> 00:00:13,350 Now let us immediately
9 00:00:13,359 –> 00:00:15,260 start with the definition.
10 00:00:15,819 –> 00:00:17,459 Of course, we will talk about
11 00:00:17,469 –> 00:00:19,280 the usual map between
12 00:00:19,309 –> 00:00:20,200 two sets.
13 00:00:20,639 –> 00:00:22,500 Let us denote the map by
14 00:00:22,510 –> 00:00:24,459 F and the two sets by
15 00:00:24,469 –> 00:00:26,120 Omega_1 and Omega_2.
16 00:00:26,870 –> 00:00:28,579 In addition, both sets
17 00:00:28,590 –> 00:00:30,549 should also comprise a Sigma
18 00:00:30,559 –> 00:00:32,150 algebra respectively.
19 00:00:32,700 –> 00:00:34,040 So let me denote the first
20 00:00:34,049 –> 00:00:35,860 Sigma algebra by A_1
21 00:00:35,969 –> 00:00:37,389 and the
22 00:00:37,400 –> 00:00:39,229 second by A_2.
23 00:00:40,560 –> 00:00:42,380 By now, you know that such
24 00:00:42,389 –> 00:00:44,189 a pair consisting of a
25 00:00:44,200 –> 00:00:45,729 set with a corresponding
26 00:00:45,740 –> 00:00:47,709 Sigma algebra is called a
27 00:00:47,720 –> 00:00:49,330 measurable space.
28 00:00:49,979 –> 00:00:51,279 And you also know that we
29 00:00:51,290 –> 00:00:52,619 need such a measurable
30 00:00:52,630 –> 00:00:54,380 space to define a
31 00:00:54,389 –> 00:00:55,290 measure on it.
32 00:00:56,200 –> 00:00:57,709 However, please note that
33 00:00:57,720 –> 00:00:59,069 we don’t need a measure here.
34 00:00:59,080 –> 00:01:00,740 We just need the sets
35 00:01:00,750 –> 00:01:02,659 and the Sigma algebras, nothing
36 00:01:02,669 –> 00:01:03,029 else.
37 00:01:03,979 –> 00:01:05,699 Now f is called
38 00:01:05,709 –> 00:01:06,709 measurable
39 00:01:07,239 –> 00:01:08,559 if the following
40 00:01:08,569 –> 00:01:10,500 holds the
41 00:01:10,510 –> 00:01:12,050 preimage of a
42 00:01:12,059 –> 00:01:13,459 set coming
43 00:01:13,470 –> 00:01:15,080 from the Sigma algebra
44 00:01:15,309 –> 00:01:16,260 A_2.
45 00:01:16,459 –> 00:01:17,629 So A_2 is an
46 00:01:17,639 –> 00:01:19,039 element in
47 00:01:19,050 –> 00:01:19,879 A_2,
48 00:01:20,709 –> 00:01:22,680 is an element
49 00:01:22,690 –> 00:01:24,240 in the sigma algebra
50 00:01:24,249 –> 00:01:25,089 A_1.
51 00:01:26,720 –> 00:01:28,370 And this holds for all
52 00:01:28,379 –> 00:01:30,220 sets from the Sigma
53 00:01:30,230 –> 00:01:31,500 algebra A_2.
54 00:01:32,569 –> 00:01:34,459 If you remember that we also
55 00:01:34,470 –> 00:01:36,239 call elements from a sigma
56 00:01:36,250 –> 00:01:37,290 algebra just
57 00:01:37,300 –> 00:01:39,089 measurable, then we can
58 00:01:39,099 –> 00:01:40,750 say a map is
59 00:01:40,760 –> 00:01:42,309 called measurable,
60 00:01:42,370 –> 00:01:44,230 if the preimages of
61 00:01:44,239 –> 00:01:46,099 measurable sets is again
62 00:01:46,110 –> 00:01:47,190 measurable.
63 00:01:47,739 –> 00:01:49,010 And of course, this could
64 00:01:49,019 –> 00:01:50,580 be confusing if there are
65 00:01:50,589 –> 00:01:51,900 a lot of different sigma
66 00:01:52,150 –> 00:01:52,989 algebras involved.
67 00:01:53,489 –> 00:01:54,779 Therefore, we would also
68 00:01:54,790 –> 00:01:56,199 include the sigma
69 00:01:56,209 –> 00:01:57,010 algebras in the
70 00:01:57,029 –> 00:01:58,680 definition of
71 00:01:58,690 –> 00:01:59,580 measurable.
72 00:01:59,589 –> 00:02:01,349 So the map f is here
73 00:02:01,360 –> 00:02:02,779 measurable with
74 00:02:02,790 –> 00:02:04,519 respect to the two sigma
75 00:02:04,529 –> 00:02:05,339 algebras.
76 00:02:07,230 –> 00:02:07,690 OK.
77 00:02:07,699 –> 00:02:09,330 So this concept seems
78 00:02:09,339 –> 00:02:11,020 meaningful because it connects
79 00:02:11,029 –> 00:02:12,860 the two sigma algebra here in
80 00:02:12,869 –> 00:02:13,660 the definition.
81 00:02:14,229 –> 00:02:15,770 However, of course, you can
82 00:02:15,779 –> 00:02:17,350 ask why do we need
83 00:02:17,360 –> 00:02:17,889 this?
84 00:02:18,100 –> 00:02:19,610 And why do we need this
85 00:02:19,619 –> 00:02:21,419 exactly in this fashion
86 00:02:21,429 –> 00:02:22,929 with the preimages?
87 00:02:23,729 –> 00:02:25,550 We can easily explain that
88 00:02:25,729 –> 00:02:27,690 when we recall why we
89 00:02:27,699 –> 00:02:29,169 do measure theory
90 00:02:29,300 –> 00:02:31,190 and how we could define
91 00:02:31,199 –> 00:02:32,550 an integral in the end.
92 00:02:33,100 –> 00:02:34,119 Therefore, let us look at
93 00:02:34,130 –> 00:02:36,110 an easy function like from
94 00:02:36,119 –> 00:02:37,649 school, so a one dimensional
95 00:02:37,660 –> 00:02:39,529 function where you can draw
96 00:02:39,539 –> 00:02:40,220 the graph.
97 00:02:41,029 –> 00:02:41,369 OK.
98 00:02:41,380 –> 00:02:42,899 So here the function has
99 00:02:42,910 –> 00:02:44,080 the value zero.
100 00:02:44,429 –> 00:02:46,270 And here I set the value
101 00:02:46,279 –> 00:02:46,759 one.
102 00:02:47,949 –> 00:02:49,270 Now the integral of this
103 00:02:49,279 –> 00:02:51,210 function should be exactly
104 00:02:51,460 –> 00:02:53,130 the volume of the
105 00:02:53,139 –> 00:02:54,770 set that is sent to one.
106 00:02:54,779 –> 00:02:56,699 So you could imagine that
107 00:02:56,710 –> 00:02:57,610 in this way,
108 00:02:58,529 –> 00:03:00,179 there’s the set and this
109 00:03:00,190 –> 00:03:01,669 is what we want to measure
110 00:03:01,889 –> 00:03:02,789 in our measure.
111 00:03:03,919 –> 00:03:05,699 So this set here on the
112 00:03:05,710 –> 00:03:06,850 xaxis is the
113 00:03:06,855 –> 00:03:09,199 preimage of the value
114 00:03:09,210 –> 00:03:09,899 1.
115 00:03:11,000 –> 00:03:12,610 So exactly this
116 00:03:12,619 –> 00:03:13,449 set here
117 00:03:14,589 –> 00:03:16,270 and what we need and have
118 00:03:16,279 –> 00:03:18,270 for the xaxis is just a
119 00:03:18,279 –> 00:03:19,250 measure space.
120 00:03:20,429 –> 00:03:21,380 So set
121 00:03:22,610 –> 00:03:24,149 Sigma algebra and
122 00:03:24,160 –> 00:03:24,740 a measure mu.
123 00:03:26,399 –> 00:03:27,979 So an abstract measure space
124 00:03:27,990 –> 00:03:28,830 is enough.
125 00:03:28,839 –> 00:03:30,059 And then we can measure
126 00:03:31,119 –> 00:03:32,759 the volume of this set here.
127 00:03:34,190 –> 00:03:35,470 Hence, for a meaningful
128 00:03:35,479 –> 00:03:37,220 integral, we want
129 00:03:37,229 –> 00:03:39,169 to measure sets of
130 00:03:39,179 –> 00:03:39,929 this form.
131 00:03:40,800 –> 00:03:42,339 However, the measure mu is
132 00:03:42,350 –> 00:03:43,750 only defined on the Sigma
133 00:03:43,759 –> 00:03:44,380 algebra.
134 00:03:44,389 –> 00:03:46,139 So what we really need
135 00:03:46,149 –> 00:03:47,830 is that this one is an
136 00:03:47,839 –> 00:03:49,740 element in the Sigma algebra
137 00:03:49,750 –> 00:03:51,100 on Omega.
138 00:03:51,139 –> 00:03:52,690 So which is here our
139 00:03:52,699 –> 00:03:53,259 A
140 00:03:55,210 –> 00:03:56,389 and there you see why we
141 00:03:56,399 –> 00:03:57,750 need the preimage in the
142 00:03:57,759 –> 00:03:59,270 definition, we want to
143 00:03:59,279 –> 00:04:01,100 guarantee that if we have
144 00:04:01,110 –> 00:04:02,990 a measurable set on the
145 00:04:03,000 –> 00:04:04,539 yaxis, we also
146 00:04:04,550 –> 00:04:06,330 get a measurable set
147 00:04:06,339 –> 00:04:07,649 on the xaxis
148 00:04:08,440 –> 00:04:09,839 and the set on the xaxis
149 00:04:09,850 –> 00:04:11,320 we get with the preimage.
150 00:04:11,330 –> 00:04:12,270 And that is what we want
151 00:04:12,279 –> 00:04:13,720 to put in the measure in
152 00:04:13,729 –> 00:04:15,339 the end or to say it
153 00:04:15,350 –> 00:04:16,858 in a few words, the
154 00:04:16,869 –> 00:04:18,600 definition of measurable
155 00:04:18,608 –> 00:04:20,048 maps solves a
156 00:04:20,059 –> 00:04:21,660 problem that will come
157 00:04:21,670 –> 00:04:22,589 later.
158 00:04:23,459 –> 00:04:23,929 OK.
159 00:04:23,940 –> 00:04:25,790 Then let’s talk about a
160 00:04:25,799 –> 00:04:26,850 few examples
161 00:04:26,859 –> 00:04:28,420 now. Then let us
162 00:04:28,429 –> 00:04:29,579 start with an
163 00:04:29,589 –> 00:04:31,320 abstract measurable
164 00:04:31,329 –> 00:04:31,709 space.
165 00:04:31,720 –> 00:04:33,119 So Omega,
166 00:04:33,130 –> 00:04:34,950 A and
167 00:04:34,959 –> 00:04:36,500 also the
168 00:04:36,850 –> 00:04:38,029 real number line
169 00:04:38,459 –> 00:04:40,000 with the Borel Sigma
170 00:04:40,359 –> 00:04:40,500 algebra.
171 00:04:43,339 –> 00:04:44,920 Now one can look at the
172 00:04:44,929 –> 00:04:46,440 common characteristic
173 00:04:46,450 –> 00:04:48,200 function. Sometimes
174 00:04:48,209 –> 00:04:49,989 also called the indicator
175 00:04:50,000 –> 00:04:50,589 function.
176 00:04:53,239 –> 00:04:54,859 And I will always
177 00:04:54,869 –> 00:04:56,609 use the greek letter Chi
178 00:04:57,220 –> 00:04:58,579 to denote this
179 00:04:58,589 –> 00:04:59,820 characteristic function.
180 00:05:00,589 –> 00:05:02,119 Namely Chi with an
181 00:05:02,130 –> 00:05:04,059 index A, where A is
182 00:05:04,070 –> 00:05:05,750 just a set in Omega.
183 00:05:06,839 –> 00:05:08,660 So the map goes from Omega
184 00:05:08,670 –> 00:05:10,010 to the real numbers
185 00:05:10,239 –> 00:05:12,220 and is defined by the
186 00:05:12,230 –> 00:05:13,739 following two cases.
187 00:05:14,510 –> 00:05:15,809 By the way, I rather should
188 00:05:15,820 –> 00:05:17,690 use a lower case omega
189 00:05:17,700 –> 00:05:19,239 to denote the variable here.
190 00:05:19,739 –> 00:05:21,619 So the two cases then are
191 00:05:21,640 –> 00:05:23,459 one or zero depending
192 00:05:23,470 –> 00:05:25,010 if our lower case omega
193 00:05:25,019 –> 00:05:26,630 comes from A
194 00:05:27,040 –> 00:05:27,730 or not.
195 00:05:28,260 –> 00:05:29,959 So omega in A is
196 00:05:29,970 –> 00:05:31,910 one and omega
197 00:05:31,920 –> 00:05:33,359 not in A
198 00:05:33,769 –> 00:05:34,529 is zero
199 00:05:35,910 –> 00:05:37,570 from this, we can now
200 00:05:37,579 –> 00:05:39,179 conclude that for
201 00:05:39,190 –> 00:05:40,440 all measurable
202 00:05:40,450 –> 00:05:42,290 sets, which means
203 00:05:42,299 –> 00:05:44,170 A in our Sigma algebra
204 00:05:44,179 –> 00:05:45,339 A, the
205 00:05:45,350 –> 00:05:47,169 characteristic function Chi_A
206 00:05:47,270 –> 00:05:48,059 is a
207 00:05:48,070 –> 00:05:49,429 measurable
208 00:05:49,440 –> 00:05:50,089 map.
209 00:05:50,720 –> 00:05:51,950 In order to prove this, we
210 00:05:51,959 –> 00:05:53,320 just have to look at all
211 00:05:53,329 –> 00:05:54,489 the preimages.
212 00:05:55,320 –> 00:05:56,700 And because we only have
213 00:05:56,709 –> 00:05:58,489 two values for the function,
214 00:05:58,500 –> 00:05:59,750 there are not so many
215 00:05:59,759 –> 00:06:01,359 preimages we could have.
216 00:06:01,510 –> 00:06:02,970 For example, if you look
217 00:06:02,980 –> 00:06:04,739 at the preimage of the
218 00:06:04,750 –> 00:06:06,579 empty set and then of
219 00:06:06,589 –> 00:06:08,230 course, you get out the empty
220 00:06:08,239 –> 00:06:09,950 set and
221 00:06:09,959 –> 00:06:11,549 the other trivial case
222 00:06:11,559 –> 00:06:13,049 would be looking what the
223 00:06:13,059 –> 00:06:14,910 preimage of the whole real
224 00:06:14,920 –> 00:06:16,589 number line is and of
225 00:06:16,600 –> 00:06:18,000 course, what you get out
226 00:06:18,010 –> 00:06:19,450 is what you put in.
227 00:06:19,459 –> 00:06:20,989 So the preimage is
228 00:06:21,000 –> 00:06:22,910 here Omega itself,
229 00:06:23,579 –> 00:06:24,910 of course, this is what we
230 00:06:24,920 –> 00:06:26,500 have for all maps of this
231 00:06:26,510 –> 00:06:27,059 form.
232 00:06:27,690 –> 00:06:29,239 So let us look at more
233 00:06:29,250 –> 00:06:30,399 interesting cases.
234 00:06:30,410 –> 00:06:31,510 So the preimage
235 00:06:31,890 –> 00:06:33,470 of for
236 00:06:33,480 –> 00:06:35,320 example, the set that contains
237 00:06:35,329 –> 00:06:36,109 only one
238 00:06:37,160 –> 00:06:38,839 and there we know the
239 00:06:38,850 –> 00:06:40,579 whole set A
240 00:06:40,589 –> 00:06:41,859 is sent to one.
241 00:06:41,869 –> 00:06:43,380 So the preimage here is
242 00:06:43,390 –> 00:06:44,820 exactly A.
243 00:06:45,109 –> 00:06:46,619 On the other hand, if we
244 00:06:46,630 –> 00:06:48,179 look at the preimage of
245 00:06:48,190 –> 00:06:49,980 the set that only contains
246 00:06:49,989 –> 00:06:51,619 zero, then we get
247 00:06:51,630 –> 00:06:53,019 out everything
248 00:06:53,029 –> 00:06:53,779 else.
249 00:06:53,790 –> 00:06:55,579 So not A. Which means
250 00:06:55,589 –> 00:06:57,339 this is the complement of
251 00:06:57,350 –> 00:06:57,589 A.
252 00:06:58,579 –> 00:07:00,440 And in fact, these are all
253 00:07:00,450 –> 00:07:01,880 the preimages we can get
254 00:07:01,890 –> 00:07:02,380 out.
255 00:07:02,390 –> 00:07:03,369 So if you choose another
256 00:07:03,380 –> 00:07:05,040 set here you get one of
257 00:07:05,049 –> 00:07:06,190 these four sets.
258 00:07:07,029 –> 00:07:08,359 And important to note here
259 00:07:08,369 –> 00:07:09,769 is of course, yeah,
260 00:07:09,779 –> 00:07:11,649 empty set, omega
261 00:07:11,890 –> 00:07:13,489 are in the sigma algebra.
262 00:07:13,970 –> 00:07:15,809 And by assumption our A is
263 00:07:15,820 –> 00:07:17,329 also the Sigma algebra.
264 00:07:17,459 –> 00:07:19,220 But then by the definition
265 00:07:19,230 –> 00:07:20,600 of the Sigma algebra, we
266 00:07:20,609 –> 00:07:22,070 also know that the
267 00:07:22,079 –> 00:07:23,619 complement is also the Sigma
268 00:07:23,630 –> 00:07:24,290 algebra.
269 00:07:24,440 –> 00:07:25,970 And please recall that is
270 00:07:25,980 –> 00:07:27,880 exactly the definition of
271 00:07:27,890 –> 00:07:29,209 a measurable map.
272 00:07:30,299 –> 00:07:32,290 So you see a characteristic
273 00:07:32,299 –> 00:07:33,480 or an indicator function
274 00:07:33,489 –> 00:07:35,290 is a typical example of a
275 00:07:35,299 –> 00:07:36,559 measurable map.
276 00:07:37,130 –> 00:07:39,109 And indeed an important one,
277 00:07:39,119 –> 00:07:40,799 because if you look at the
278 00:07:40,809 –> 00:07:42,429 picture here again, these
279 00:07:42,440 –> 00:07:44,010 are maps that we could
280 00:07:44,019 –> 00:07:45,970 easily integrate if we have
281 00:07:45,980 –> 00:07:47,019 a notion of integration.
282 00:07:47,029 –> 00:07:48,420 So this is what we will do
283 00:07:48,429 –> 00:07:49,750 in the next videos later.
284 00:07:50,640 –> 00:07:52,059 However, here we should first
285 00:07:52,070 –> 00:07:53,440 continue with the next
286 00:07:53,450 –> 00:07:54,220 example.
287 00:07:54,609 –> 00:07:56,410 Let us choose here three
288 00:07:56,420 –> 00:07:58,119 measurable spaces.
289 00:07:58,700 –> 00:08:00,149 We do this because I want
290 00:08:00,160 –> 00:08:02,059 to look at compositions of
291 00:08:02,070 –> 00:08:03,380 measurable maps.
292 00:08:04,089 –> 00:08:05,459 In other words, we start
293 00:08:05,470 –> 00:08:07,309 with the set Omega_1 go
294 00:08:07,320 –> 00:08:08,579 in Omega_2
295 00:08:08,589 –> 00:08:10,440 and then in Omega_3.
296 00:08:11,359 –> 00:08:12,660 let us then call the map
297 00:08:12,670 –> 00:08:14,239 here by f
298 00:08:14,609 –> 00:08:16,160 and this one should be the
299 00:08:16,170 –> 00:08:17,220 map g.
300 00:08:17,440 –> 00:08:18,820 So and then we can do
301 00:08:18,829 –> 00:08:20,750 the composition. Which
302 00:08:20,760 –> 00:08:22,450 means a map going
303 00:08:22,459 –> 00:08:23,950 from Omega_1
304 00:08:23,959 –> 00:08:25,029 directly to
305 00:08:25,040 –> 00:08:26,209 Omega_3.
306 00:08:26,230 –> 00:08:27,869 And this is then defined
307 00:08:27,880 –> 00:08:29,660 as g after
308 00:08:29,670 –> 00:08:30,329 f.
309 00:08:30,339 –> 00:08:31,910 So the composition of both
310 00:08:31,920 –> 00:08:33,758 maps. The claim is
311 00:08:33,768 –> 00:08:35,679 now, if we put in
312 00:08:35,688 –> 00:08:37,558 two measurable maps,
313 00:08:37,789 –> 00:08:39,419 then also the composition
314 00:08:39,429 –> 00:08:40,597 is measurable.
315 00:08:41,148 –> 00:08:42,328 So in short, this
316 00:08:42,337 –> 00:08:44,239 implies g
317 00:08:44,689 –> 00:08:46,018 after f
318 00:08:46,028 –> 00:08:47,059 measurable.
319 00:08:47,919 –> 00:08:49,619 Indeed, this is very easy
320 00:08:49,630 –> 00:08:51,559 to see if you look again
321 00:08:51,570 –> 00:08:52,739 at the preimages.
322 00:08:53,219 –> 00:08:54,820 So the preimage of
323 00:08:54,830 –> 00:08:56,659 g composed with f
324 00:08:57,059 –> 00:08:58,650 of a set
325 00:08:58,659 –> 00:09:00,419 coming from the Sigma
326 00:09:00,429 –> 00:09:01,669 algebra A_3.
327 00:09:01,679 –> 00:09:03,090 So let’s call it A_3
328 00:09:03,859 –> 00:09:05,400 is equal to
329 00:09:05,690 –> 00:09:07,039 and now we can go
330 00:09:07,090 –> 00:09:08,380 stepwise.
331 00:09:08,390 –> 00:09:10,210 So we have
332 00:09:10,219 –> 00:09:11,679 the preimage of g
333 00:09:11,690 –> 00:09:12,480 first
334 00:09:12,590 –> 00:09:13,900 of A_3.
335 00:09:13,909 –> 00:09:15,460 So this step and then the
336 00:09:15,469 –> 00:09:16,479 preimage of this.
337 00:09:17,989 –> 00:09:19,890 So this equality holds for
338 00:09:19,900 –> 00:09:20,539 all maps.
339 00:09:20,549 –> 00:09:21,849 But now we can put in what
340 00:09:21,859 –> 00:09:23,530 we know of f and
341 00:09:23,539 –> 00:09:25,440 g. Maybe first we know
342 00:09:25,450 –> 00:09:27,250 that g is measurable.
343 00:09:27,530 –> 00:09:29,510 So we know this one is
344 00:09:29,520 –> 00:09:31,299 an element in the Sigma
345 00:09:31,309 –> 00:09:32,909 algebra A_2.
346 00:09:33,640 –> 00:09:35,359 But now we know that f is
347 00:09:35,369 –> 00:09:36,900 also measurable.
348 00:09:36,929 –> 00:09:38,130 So we know that the whole
349 00:09:38,140 –> 00:09:39,659 thing here now
350 00:09:39,669 –> 00:09:41,380 lies in the Sigma algebra
351 00:09:41,390 –> 00:09:43,289 A_1 or in other
352 00:09:43,299 –> 00:09:44,820 words, does not matter
353 00:09:44,830 –> 00:09:46,609 which element from the sigma
354 00:09:46,619 –> 00:09:47,919 algebra A_3
355 00:09:47,929 –> 00:09:49,180 you choose here the
356 00:09:49,219 –> 00:09:51,380 preimage always lies in
357 00:09:51,390 –> 00:09:53,020 the sigma algebra A_1.
358 00:09:53,789 –> 00:09:55,390 And this is indeed what
359 00:09:55,400 –> 00:09:56,869 measurable for maps
360 00:09:56,880 –> 00:09:57,450 means.
361 00:09:58,559 –> 00:10:00,500 Well, let’s close this video
362 00:10:00,510 –> 00:10:02,059 with some important facts
363 00:10:02,070 –> 00:10:03,419 about measurable
364 00:10:03,429 –> 00:10:04,229 functions.
365 00:10:04,690 –> 00:10:06,210 And when I say function,
366 00:10:06,219 –> 00:10:08,119 I just mean a map that
367 00:10:08,130 –> 00:10:09,570 maps into the real number
368 00:10:09,580 –> 00:10:11,419 line. This means
369 00:10:11,429 –> 00:10:13,059 that we now have two
370 00:10:13,070 –> 00:10:14,849 measurable spaces.
371 00:10:14,989 –> 00:10:16,739 And one of them is just the
372 00:10:16,750 –> 00:10:18,099 real number line with the
373 00:10:18,109 –> 00:10:19,520 Borel Sigma algebra.
374 00:10:22,130 –> 00:10:23,950 Now let us look at two maps.
375 00:10:23,960 –> 00:10:25,849 So just functions.
376 00:10:25,859 –> 00:10:27,780 Let’s start with omega
377 00:10:28,070 –> 00:10:29,909 and land in the real
378 00:10:29,919 –> 00:10:30,700 number line.
379 00:10:31,159 –> 00:10:32,349 And of course, they should
380 00:10:32,359 –> 00:10:34,030 now be measurable
381 00:10:34,039 –> 00:10:35,940 with respect to these two
382 00:10:35,950 –> 00:10:37,179 Sigma algebras
383 00:10:38,049 –> 00:10:39,260 because we have the whole
384 00:10:39,270 –> 00:10:40,940 structure of the real
385 00:10:40,950 –> 00:10:42,630 numbers, we can now
386 00:10:42,640 –> 00:10:44,260 look at combinations of these
387 00:10:44,270 –> 00:10:45,070 two functions.
388 00:10:45,820 –> 00:10:47,609 For example, f +
389 00:10:47,619 –> 00:10:49,159 g and f
390 00:10:49,169 –> 00:10:51,039
 g again will
391 00:10:51,049 –> 00:10:52,250 define functions.
392 00:10:52,359 –> 00:10:53,900 And the result here is they
393 00:10:53,909 –> 00:10:55,710 are still measurable
394 00:10:56,409 –> 00:10:57,599 in the same manner.
395 00:10:57,609 –> 00:10:59,440 We can look at the multiplication
396 00:10:59,450 –> 00:11:01,140 of the two functions just
397 00:11:01,150 –> 00:11:03,109 by multiplying the values
398 00:11:03,119 –> 00:11:04,289 and defining the function.
399 00:11:04,299 –> 00:11:05,380 So as always,
400 00:11:06,020 –> 00:11:07,359 and this new function is
401 00:11:07,369 –> 00:11:09,340 then again measurable
402 00:11:10,409 –> 00:11:12,090 and also very important
403 00:11:12,159 –> 00:11:13,669 if you ignore
404 00:11:13,679 –> 00:11:15,039 negative values, so make
405 00:11:15,049 –> 00:11:15,969 them positive.
406 00:11:16,200 –> 00:11:17,330 So you get out the absolute
407 00:11:17,340 –> 00:11:18,669 value as a function.
408 00:11:18,760 –> 00:11:20,510 This one is also
409 00:11:20,520 –> 00:11:21,510 measurable.
410 00:11:22,429 –> 00:11:24,190 There are also other important
411 00:11:24,200 –> 00:11:26,099 combinations one considers
412 00:11:26,299 –> 00:11:27,869 and we will later see if
413 00:11:27,880 –> 00:11:28,650 we need them.
414 00:11:29,289 –> 00:11:31,049 However, the idea is always
415 00:11:31,059 –> 00:11:31,789 the same.
416 00:11:31,799 –> 00:11:33,080 So you can prove this in
417 00:11:33,090 –> 00:11:34,960 general and use it the whole
418 00:11:34,969 –> 00:11:36,640 time in measure theory.
419 00:11:37,510 –> 00:11:38,030 OK.
420 00:11:38,039 –> 00:11:39,229 I think that’s good enough
421 00:11:39,239 –> 00:11:40,030 for today.
422 00:11:40,039 –> 00:11:42,030 So now you learned what a
423 00:11:42,039 –> 00:11:43,710 measurable map is
424 00:11:44,669 –> 00:11:46,609 and as I promised before,
425 00:11:46,859 –> 00:11:48,729 you will need this definition
426 00:11:48,739 –> 00:11:49,929 when we consider
427 00:11:49,940 –> 00:11:51,270 integration theory.
428 00:11:52,039 –> 00:11:53,080 And that is what you will
429 00:11:53,090 –> 00:11:54,619 see in the next video
430 00:11:54,840 –> 00:11:56,559 when we indeed start
431 00:11:56,570 –> 00:11:57,909 with the Lebesgue integral
432 00:11:59,200 –> 00:12:00,859 then see you next time
433 00:12:00,869 –> 00:12:01,239 there.
434 00:12:01,510 –> 00:12:02,130 Bye.

Quiz Content
Q1: Let $(\Omega_1, \mathcal{A}_1)$ and $(\Omega_2, \mathcal{A}_2)$ be measurable spaces. What is the correct definition for a map $f: \Omega_1 \rightarrow \Omega_2$ being a measurable map?
A1: $f(A_1) = f(A_2)$ for all $A_1 \in \mathcal{A}_1$ and $A_2 \in \mathcal{A}_2$.
A2: $f(A_1) \in \mathcal{A}_2 $ for all $A_1 \in \mathcal{A}_1$.
A3: $f^{1}(A_1) \in \mathcal{A}_2 $ for all $A_1 \in \mathcal{A}_1$.
A4: $f^{1}(A_2) \in \mathcal{A}_1 $ for all $A_2 \in \mathcal{A}_2$.
A5: $f(A_2) \in \mathcal{A}_1 $ for all $A_2 \in \mathcal{A}_2$.
Q2: Let $(\Omega, \mathcal{A})$ be a measurable space. In which cases is the indicator function $\chi_A : \Omega \rightarrow \mathbb{R}$ a measurable map?
A1: Always!
A2: If $A \in \mathcal{A}$.
A3: Never!
Q3: Is the composition of measurable maps also measurable?
A1: Yes.
A2: No.
A3: Only in special cases.
Q4: Let $(\mathbb{R}, \mathcal{B}(\mathbb{R}) )$ be the measurable space given by the Borel sigma algebra. Is the function $\chi_{[0,1]} + \chi_{{ 5 }}$ measurable?
A1: Yes, it is.
A2: No, it isn’t.
A3: One needs more information.
Q5: Let $(\mathbb{R}, \mathcal{B}(\mathbb{R}) )$ be the measurable space given by the Borel sigma algebra. Is the function $\chi_{[0,1]} \cdot \chi_{ [0,\infty) }$ measurable?
A1: Yes, it is.
A2: No, it isn’t.
A3: One needs more information.