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Title: Radon-Nikodym theorem and Lebesgue’s decomposition theorem
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Series: Measure Theory
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YouTube-Title: Measure Theory 14 | Radon-Nikodym theorem and Lebesgue’s decomposition theorem
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Subtitle in English
1 00:00:01,120 –> 00:00:02,705 Hello and welcome to a new
2 00:00:02,745 –> 00:00:04,685 video about measure theory.
3 00:00:05,425 –> 00:00:07,081 First, let me thank all the
4 00:00:07,113 –> 00:00:08,969 nice people who support this
5 00:00:09,017 –> 00:00:10,565 channel on Steady.
6 00:00:11,025 –> 00:00:12,777 Today’s video is about
7 00:00:12,801 –> 00:00:14,617 Carathéodory’s extension
8 00:00:14,681 –> 00:00:15,525 theorem.
9 00:00:16,345 –> 00:00:17,881 Okay, so this is a video
10 00:00:17,953 –> 00:00:19,537 in my measure theory
11 00:00:19,641 –> 00:00:20,169 series.
12 00:00:20,297 –> 00:00:21,977 But you don’t have to know
13 00:00:22,121 –> 00:00:23,673 all the videos there
14 00:00:23,849 –> 00:00:25,297 for understanding
15 00:00:25,361 –> 00:00:27,173 Carathéodory’s extension theorem.
16 00:00:27,289 –> 00:00:28,717 It’s sufficient when you
17 00:00:28,781 –> 00:00:30,717 know what a measure is
18 00:00:30,781 –> 00:00:31,949 and what a sigma algebra
19 00:00:31,997 –> 00:00:32,585 is.
20 00:00:33,005 –> 00:00:34,421 All the other notions we
21 00:00:34,453 –> 00:00:36,293 need I will explain here
22 00:00:36,349 –> 00:00:36,945 today.
23 00:00:37,685 –> 00:00:39,429 So let’s start immediately
24 00:00:39,517 –> 00:00:40,745 with the theorem.
25 00:00:41,165 –> 00:00:42,605 As always, we have a set
26 00:00:42,685 –> 00:00:44,573 X and we also look at
27 00:00:44,669 –> 00:00:46,389 subsets of the set
28 00:00:46,437 –> 00:00:46,741 X.
29 00:00:46,813 –> 00:00:48,245 So we look at a collection
30 00:00:48,285 –> 00:00:50,173 of these subsets and we always
31 00:00:50,229 –> 00:00:51,845 call this by this curved
32 00:00:51,925 –> 00:00:52,545 A.
33 00:00:53,815 –> 00:00:55,671 In general, this is not
34 00:00:55,743 –> 00:00:57,655 a sigma algebra now, but
35 00:00:57,735 –> 00:00:59,207 only a so called
36 00:00:59,311 –> 00:01:00,395 semi ring.
37 00:01:01,775 –> 00:01:03,487 To be more specific, I will
38 00:01:03,511 –> 00:01:05,199 call it a semi ring of
39 00:01:05,287 –> 00:01:05,983 sets.
40 00:01:06,119 –> 00:01:07,335 Then you know that we are
41 00:01:07,375 –> 00:01:08,367 talking about a collection
42 00:01:08,391 –> 00:01:09,315 of sets.
43 00:01:09,775 –> 00:01:11,543 The explicit definition of
44 00:01:11,559 –> 00:01:13,175 a semi ring I will give you
45 00:01:13,215 –> 00:01:13,795 later.
46 00:01:14,495 –> 00:01:15,751 However, in the same way
47 00:01:15,783 –> 00:01:17,247 as for the sigma algebras,
48 00:01:17,351 –> 00:01:19,267 we also have a map
49 00:01:19,331 –> 00:01:20,695 we call mu
50 00:01:21,075 –> 00:01:22,643 defined on
51 00:01:22,779 –> 00:01:24,707 the whole collection of subsets.
52 00:01:24,811 –> 00:01:26,331 So on this semi ring
53 00:01:26,363 –> 00:01:28,219 here, and it maps into
54 00:01:28,267 –> 00:01:29,635 the non negative real
55 00:01:29,715 –> 00:01:31,075 numbers where we
56 00:01:31,155 –> 00:01:32,899 include infinity
57 00:01:33,027 –> 00:01:34,135 as a symbol.
58 00:01:34,595 –> 00:01:34,971 Okay.
59 00:01:35,003 –> 00:01:36,611 In the case that A is not
60 00:01:36,643 –> 00:01:38,531 a sigma algebra, this can’t
61 00:01:38,563 –> 00:01:39,815 be a measure.
62 00:01:40,155 –> 00:01:41,371 And therefore there is another
63 00:01:41,443 –> 00:01:42,595 name for such a thing.
64 00:01:42,675 –> 00:01:44,027 We call it a pre
65 00:01:44,091 –> 00:01:44,815 measure.
66 00:01:45,245 –> 00:01:46,917 Of course, this name tells
67 00:01:46,941 –> 00:01:47,845 you a lot.
68 00:01:48,005 –> 00:01:49,829 It’s a map that you
69 00:01:49,877 –> 00:01:51,685 define before you can
70 00:01:51,725 –> 00:01:53,585 define a real measure.
71 00:01:53,885 –> 00:01:55,525 Okay, I will tell you more
72 00:01:55,565 –> 00:01:56,589 about this later.
73 00:01:56,717 –> 00:01:58,541 First let us focus on the
74 00:01:58,573 –> 00:01:59,585 theorem here.
75 00:02:00,005 –> 00:02:01,265 The first claim.
76 00:02:01,685 –> 00:02:03,485 Obviously you have an
77 00:02:03,525 –> 00:02:04,425 extension,
78 00:02:05,085 –> 00:02:06,505 then mu
79 00:02:06,845 –> 00:02:08,021 has an
80 00:02:08,093 –> 00:02:08,985 extension
81 00:02:10,894 –> 00:02:11,910 and we call this
82 00:02:11,982 –> 00:02:13,774 extension by
83 00:02:13,934 –> 00:02:15,062 mu tilde.
84 00:02:15,158 –> 00:02:16,774 And indeed this is an
85 00:02:16,814 –> 00:02:18,790 ordinary measure as we already
86 00:02:18,862 –> 00:02:19,674 know it.
87 00:02:20,334 –> 00:02:21,934 Therefore it really needs
88 00:02:22,014 –> 00:02:23,878 a sigma algebra as
89 00:02:23,926 –> 00:02:24,846 its domain.
90 00:02:24,990 –> 00:02:26,278 So we call this
91 00:02:26,406 –> 00:02:27,734 sigma of
92 00:02:27,814 –> 00:02:28,434 A.
93 00:02:28,894 –> 00:02:30,430 Please remember this one
94 00:02:30,462 –> 00:02:32,214 is just a short notation
95 00:02:32,334 –> 00:02:34,190 for the sigma algebra that
96 00:02:34,222 –> 00:02:35,678 is generated by the
97 00:02:35,726 –> 00:02:37,407 subsets that are elements
98 00:02:37,471 –> 00:02:38,315 in A.
99 00:02:38,775 –> 00:02:39,935 And of course we have to
100 00:02:39,975 –> 00:02:41,463 land again in our
101 00:02:41,519 –> 00:02:43,431 interval 0 to infinity,
102 00:02:43,503 –> 00:02:45,195 where we include infinity.
103 00:02:45,975 –> 00:02:47,583 Hence the claim of the theorem
104 00:02:47,639 –> 00:02:49,415 is if you have these
105 00:02:49,495 –> 00:02:51,311 things, you get out an
106 00:02:51,343 –> 00:02:52,535 actual measure.
107 00:02:52,695 –> 00:02:54,487 And extension now just
108 00:02:54,551 –> 00:02:56,391 means that if you look at
109 00:02:56,463 –> 00:02:58,255 the original subsets in
110 00:02:58,295 –> 00:03:00,239 A, it does not matter if
111 00:03:00,247 –> 00:03:01,687 you measure it with mu or
112 00:03:01,711 –> 00:03:02,755 mu tilde.
113 00:03:04,835 –> 00:03:06,731 So the new measure is
114 00:03:06,763 –> 00:03:08,547 indeed bigger or better
115 00:03:08,691 –> 00:03:10,375 than the pre measure before,
116 00:03:10,715 –> 00:03:12,299 but you don’t change the
117 00:03:12,347 –> 00:03:13,695 original values.
118 00:03:14,555 –> 00:03:16,539 Okay, so this was part
119 00:03:16,627 –> 00:03:18,603 a, which tells you that
120 00:03:18,659 –> 00:03:20,403 there is an extension.
121 00:03:20,539 –> 00:03:22,259 So this is an existence
122 00:03:22,347 –> 00:03:22,975 result.
123 00:03:23,275 –> 00:03:24,651 However it would be also
124 00:03:24,723 –> 00:03:26,227 Nice to have a uniqueness
125 00:03:26,291 –> 00:03:27,579 result as well.
126 00:03:27,747 –> 00:03:28,819 And this is what we will
127 00:03:28,867 –> 00:03:30,121 have in part b.
128 00:03:30,223 –> 00:03:32,221 Now however, we
129 00:03:32,253 –> 00:03:34,157 need some additional condition
130 00:03:34,221 –> 00:03:34,825 here.
131 00:03:35,245 –> 00:03:36,789 This condition is often
132 00:03:36,877 –> 00:03:37,997 called sigma
133 00:03:38,101 –> 00:03:39,061 finite.
134 00:03:39,253 –> 00:03:41,157 It means that you find a
135 00:03:41,221 –> 00:03:43,149 sequence of subsets,
136 00:03:43,317 –> 00:03:45,305 let’s call them SJ
137 00:03:46,125 –> 00:03:47,157 such that
138 00:03:47,301 –> 00:03:49,269 SJ is in our
139 00:03:49,317 –> 00:03:50,301 semi ring.
140 00:03:50,453 –> 00:03:51,585 So in A
141 00:03:52,245 –> 00:03:54,157 they should also cover the
142 00:03:54,181 –> 00:03:54,981 whole set.
143 00:03:55,093 –> 00:03:56,305 So the union
144 00:03:56,835 –> 00:03:58,435 J starting with one
145 00:03:58,595 –> 00:03:59,615 going to
146 00:03:59,915 –> 00:04:00,975 infinity
147 00:04:03,035 –> 00:04:04,563 is equal to
148 00:04:04,619 –> 00:04:05,215 X.
149 00:04:06,355 –> 00:04:08,139 And in addition, all the
150 00:04:08,187 –> 00:04:09,603 sets should have
151 00:04:09,699 –> 00:04:11,535 finite pre measure.
152 00:04:11,835 –> 00:04:13,379 This means that if we put
153 00:04:13,427 –> 00:04:15,403 the set into the function
154 00:04:15,459 –> 00:04:17,307 mu, we get out a
155 00:04:17,331 –> 00:04:18,803 finite number, so not
156 00:04:18,859 –> 00:04:20,055 infinity here.
157 00:04:21,964 –> 00:04:23,820 Now, if we have all these
158 00:04:23,892 –> 00:04:25,644 things, then the
159 00:04:25,684 –> 00:04:27,308 conclusion is that this
160 00:04:27,356 –> 00:04:28,916 extension is also
161 00:04:29,020 –> 00:04:29,784 unique.
162 00:04:30,164 –> 00:04:31,652 Now this means that we can
163 00:04:31,708 –> 00:04:33,564 summarize Carathéodory’s
164 00:04:33,604 –> 00:04:35,104 theorem in the following.
165 00:04:35,724 –> 00:04:37,268 If we have a pre
166 00:04:37,316 –> 00:04:38,924 measure that fulfills
167 00:04:39,004 –> 00:04:40,580 this condition here,
168 00:04:40,732 –> 00:04:42,476 then there is exactly
169 00:04:42,620 –> 00:04:44,132 one extension
170 00:04:44,268 –> 00:04:46,068 to the sigma
171 00:04:46,116 –> 00:04:47,706 algebra that is generated
172 00:04:47,860 –> 00:04:49,595 by this semi ring.
173 00:04:50,175 –> 00:04:51,911 So exactly one
174 00:04:52,023 –> 00:04:52,755 measure.
175 00:04:53,295 –> 00:04:54,871 Moreover, this property
176 00:04:54,943 –> 00:04:56,735 then also holds for
177 00:04:56,775 –> 00:04:58,155 our mu tilde.
178 00:04:58,615 –> 00:05:00,431 Mu tilde is also
179 00:05:00,543 –> 00:05:01,875 sigma finite.
180 00:05:02,375 –> 00:05:04,007 If you never heard this notion
181 00:05:04,071 –> 00:05:05,407 sigma finite, that’s not
182 00:05:05,431 –> 00:05:06,223 a problem.
183 00:05:06,399 –> 00:05:08,271 Definition is here and you
184 00:05:08,303 –> 00:05:09,767 should recognize it’s a
185 00:05:09,791 –> 00:05:11,775 generalization of a finite
186 00:05:11,855 –> 00:05:12,535 measure.
187 00:05:12,695 –> 00:05:14,569 For a finite measure, the
188 00:05:14,617 –> 00:05:16,193 measure of X would be
189 00:05:16,249 –> 00:05:18,073 finite, and this then
190 00:05:18,209 –> 00:05:19,405 still holds.
191 00:05:19,705 –> 00:05:21,281 But if it’s not finite,
192 00:05:21,393 –> 00:05:23,065 then sigma finite
193 00:05:23,145 –> 00:05:24,761 then tells you you can
194 00:05:24,833 –> 00:05:26,625 approximate the whole
195 00:05:26,705 –> 00:05:28,265 set with finite
196 00:05:28,345 –> 00:05:29,765 measured sets
197 00:05:30,705 –> 00:05:32,561 and you only need countable
198 00:05:32,633 –> 00:05:33,805 many of them.
199 00:05:34,785 –> 00:05:36,321 However, don’t worry, you
200 00:05:36,353 –> 00:05:37,857 will see in other videos
201 00:05:37,961 –> 00:05:39,441 why this will be important
202 00:05:39,513 –> 00:05:40,365 in future.
203 00:05:41,095 –> 00:05:42,271 Well, there you have it.
204 00:05:42,343 –> 00:05:43,935 This is Carathéodory’s
205 00:05:43,975 –> 00:05:45,515 extension theorem.
206 00:05:45,855 –> 00:05:47,583 There may be other variants
207 00:05:47,639 –> 00:05:49,415 of this theorem, but this
208 00:05:49,455 –> 00:05:50,983 version is the one I
209 00:05:51,039 –> 00:05:52,195 personally prefer.
210 00:05:53,415 –> 00:05:55,263 Now, as promised, I give
211 00:05:55,319 –> 00:05:57,175 you all the explanations
212 00:05:57,295 –> 00:05:58,527 you need for the notions
213 00:05:58,591 –> 00:05:59,195 here.
214 00:05:59,575 –> 00:06:01,287 After this I give you
215 00:06:01,351 –> 00:06:03,055 an example and also
216 00:06:03,135 –> 00:06:04,415 an application of the
217 00:06:04,455 –> 00:06:05,989 theorem.
218 00:06:06,067 –> 00:06:07,681 But first, let’s start with
219 00:06:07,713 –> 00:06:08,885 the semi ring.
220 00:06:11,025 –> 00:06:12,617 I called it a semi ring
221 00:06:12,681 –> 00:06:13,745 of sets.
222 00:06:13,865 –> 00:06:15,777 To remember you that we
223 00:06:15,841 –> 00:06:17,681 consider a subset
224 00:06:17,753 –> 00:06:19,725 in the power set of X.
225 00:06:21,465 –> 00:06:23,385 Now we need three properties
226 00:06:23,505 –> 00:06:24,889 such that we can call it
227 00:06:24,937 –> 00:06:26,045 a semiring.
228 00:06:26,745 –> 00:06:28,521 The first property is that
229 00:06:28,553 –> 00:06:30,281 the empty set is in
230 00:06:30,313 –> 00:06:32,235 all cases in the
231 00:06:32,275 –> 00:06:33,015 collection.
232 00:06:33,915 –> 00:06:35,235 If you remember the sigma
233 00:06:35,275 –> 00:06:37,147 algebra, you recognize this
234 00:06:37,171 –> 00:06:38,987 is the same thing we
235 00:06:39,011 –> 00:06:40,539 want from a sigma algebra
236 00:06:40,627 –> 00:06:41,683 at the beginning.
237 00:06:41,859 –> 00:06:43,475 The second property is now
238 00:06:43,555 –> 00:06:44,923 that we can form
239 00:06:45,059 –> 00:06:46,215 intersections.
240 00:06:46,515 –> 00:06:48,195 This means that the intersection
241 00:06:48,235 –> 00:06:49,779 of two sets in A
242 00:06:49,907 –> 00:06:51,539 lies also in
243 00:06:51,587 –> 00:06:52,175 A.
244 00:06:52,715 –> 00:06:54,467 Please also remember this
245 00:06:54,491 –> 00:06:56,419 is indeed satisfied for sigma
246 00:06:56,467 –> 00:06:57,155 algebras.
247 00:06:57,275 –> 00:06:58,475 It’s not in the definition,
248 00:06:58,595 –> 00:06:59,811 but of course it follows
249 00:06:59,843 –> 00:07:00,975 from the definition.
250 00:07:01,515 –> 00:07:02,811 However, please note that
251 00:07:02,843 –> 00:07:04,555 we don’t say anything about
252 00:07:04,635 –> 00:07:06,067 complements or unions
253 00:07:06,131 –> 00:07:06,643 here.
254 00:07:06,779 –> 00:07:08,475 So Indeed, we weaken the
255 00:07:08,515 –> 00:07:10,291 notion of the sigma algebra
256 00:07:10,403 –> 00:07:11,691 with the notion of a semi
257 00:07:11,723 –> 00:07:12,451 ring.
258 00:07:12,643 –> 00:07:14,323 But we have a third property
259 00:07:14,459 –> 00:07:15,843 where you will see a little
260 00:07:15,899 –> 00:07:17,091 bit of the unions and
261 00:07:17,123 –> 00:07:17,935 complements.
262 00:07:18,555 –> 00:07:20,507 It tells you that for arbitrary
263 00:07:20,571 –> 00:07:21,855 elements AB
264 00:07:22,395 –> 00:07:24,379 in A, you can
265 00:07:24,427 –> 00:07:26,091 look at the difference
266 00:07:26,243 –> 00:07:27,535 of the sets.
267 00:07:28,475 –> 00:07:30,131 This one here is not the
268 00:07:30,163 –> 00:07:31,771 complement because we don’t
269 00:07:31,803 –> 00:07:33,755 know if the set X is
270 00:07:33,795 –> 00:07:35,123 inside our semi ring.
271 00:07:35,219 –> 00:07:36,395 This does not follow from
272 00:07:36,435 –> 00:07:37,855 the two properties here.
273 00:07:38,355 –> 00:07:39,923 Nevertheless, we don’t want
274 00:07:39,979 –> 00:07:41,811 that this difference is again
275 00:07:41,883 –> 00:07:42,611 in A.
276 00:07:42,723 –> 00:07:44,331 We just want that we can
277 00:07:44,363 –> 00:07:46,339 find a union of sets of
278 00:07:46,387 –> 00:07:48,011 A that is given
279 00:07:48,123 –> 00:07:49,717 as the difference here.
280 00:07:49,891 –> 00:07:51,481 More concretely, this means
281 00:07:51,593 –> 00:07:53,385 there are pairwise
282 00:07:53,545 –> 00:07:54,565 disjoint
283 00:07:56,665 –> 00:07:58,457 sets and let’s call
284 00:07:58,481 –> 00:08:00,361 them S1 till
285 00:08:00,433 –> 00:08:02,169 SN, so finitely
286 00:08:02,217 –> 00:08:02,805 many.
287 00:08:03,385 –> 00:08:05,145 And they should lie in our
288 00:08:05,185 –> 00:08:07,113 semi ring A such
289 00:08:07,169 –> 00:08:08,913 that we can write this
290 00:08:08,969 –> 00:08:10,849 difference as the
291 00:08:10,937 –> 00:08:12,721 union of these
292 00:08:12,833 –> 00:08:14,065 pairwise disjoint
293 00:08:14,145 –> 00:08:14,885 sets.
294 00:08:16,235 –> 00:08:17,571 And this is then what we
295 00:08:17,603 –> 00:08:19,403 call a semi ring of
296 00:08:19,459 –> 00:08:20,215 sets.
297 00:08:20,755 –> 00:08:21,971 When you see this for the
298 00:08:22,003 –> 00:08:23,923 first time, it may look a
299 00:08:23,939 –> 00:08:25,451 little bit strange, but you
300 00:08:25,483 –> 00:08:26,947 will see later that this
301 00:08:26,971 –> 00:08:28,935 is exactly what we need
302 00:08:29,355 –> 00:08:30,843 for getting some intuition
303 00:08:30,939 –> 00:08:31,627 behind.
304 00:08:31,811 –> 00:08:32,443 Maybe.
305 00:08:32,539 –> 00:08:34,371 Let’s look at an example
306 00:08:34,563 –> 00:08:36,499 and I call this one the
307 00:08:36,547 –> 00:08:38,255 most important example.
308 00:08:39,075 –> 00:08:40,923 This one is an example in
309 00:08:40,979 –> 00:08:41,459 R.
310 00:08:41,587 –> 00:08:43,455 So the real number line
311 00:08:43,885 –> 00:08:45,797 and we look at the finite
312 00:08:45,901 –> 00:08:47,821 intervals given by
313 00:08:47,933 –> 00:08:49,773 ab, where A
314 00:08:49,829 –> 00:08:51,453 is inside the interval but
315 00:08:51,549 –> 00:08:53,065 B is not.
316 00:08:54,045 –> 00:08:55,701 And for having a good interval
317 00:08:55,773 –> 00:08:57,733 I want A less or
318 00:08:57,789 –> 00:08:59,345 equal than B.
319 00:09:00,525 –> 00:09:02,509 Obviously this is not
320 00:09:02,557 –> 00:09:03,945 a sigma algebra.
321 00:09:04,565 –> 00:09:06,469 You can see this immediately
322 00:09:06,597 –> 00:09:08,237 because only the finite
323 00:09:08,301 –> 00:09:10,195 intervals are A.
324 00:09:11,135 –> 00:09:12,911 In particular, R
325 00:09:12,983 –> 00:09:14,831 itself is not in
326 00:09:14,863 –> 00:09:15,435 A.
327 00:09:17,415 –> 00:09:19,343 But we have to note
328 00:09:19,479 –> 00:09:21,191 that A generates a
329 00:09:21,223 –> 00:09:23,047 well known sigma algebra
330 00:09:23,231 –> 00:09:24,887 and this one is the Borel
331 00:09:24,951 –> 00:09:25,991 sigma algebra.
332 00:09:26,103 –> 00:09:27,863 So we can write in short,
333 00:09:27,999 –> 00:09:29,759 sigma of A
334 00:09:29,927 –> 00:09:31,575 is equal to
335 00:09:31,695 –> 00:09:33,519 Borel sigma algebra
336 00:09:33,647 –> 00:09:34,635 of R.
337 00:09:35,345 –> 00:09:36,897 This means that if we want
338 00:09:36,921 –> 00:09:38,305 to describe the Borel sigma
339 00:09:38,345 –> 00:09:40,065 algebra, it is sufficient
340 00:09:40,145 –> 00:09:41,905 to use these finite
341 00:09:41,985 –> 00:09:42,725 intervals.
342 00:09:43,185 –> 00:09:44,873 And the beautiful thing now
343 00:09:44,929 –> 00:09:46,921 is they form a semi
344 00:09:46,953 –> 00:09:48,665 ring, which
345 00:09:48,705 –> 00:09:50,401 means they can help us with
346 00:09:50,433 –> 00:09:52,233 Carathéodory’s extension
347 00:09:52,289 –> 00:09:53,125 theorem.
348 00:09:53,625 –> 00:09:55,345 Ok, then let’s check that.
349 00:09:55,425 –> 00:09:56,753 So let’s check the three
350 00:09:56,809 –> 00:09:57,565 properties.
351 00:09:58,185 –> 00:09:59,849 Of course the empty set
352 00:09:59,937 –> 00:10:01,847 is not a problem, it’s
353 00:10:01,871 –> 00:10:03,279 formally given in this
354 00:10:03,327 –> 00:10:04,935 definition, but of course
355 00:10:04,975 –> 00:10:06,455 we always could put it in.
356 00:10:06,535 –> 00:10:08,503 But here, for a equals b,
357 00:10:08,599 –> 00:10:09,935 we have the empty set here
358 00:10:09,975 –> 00:10:10,951 because there’s no point
359 00:10:10,983 –> 00:10:12,035 in the interval.
360 00:10:12,695 –> 00 :10:14,247 So let’s look at the second
361 00:10:14,311 –> 00:10:15,271 property here.
362 00:10:15,303 –> 00:10:17,159 It’s about the intersection.
363 00:10:17,287 –> 00:10:19,215 So let’s choose two intervals,
364 00:10:19,335 –> 00:10:20,315 maybe ab
365 00:10:21,015 –> 00:10:22,119 and also
366 00:10:22,287 –> 00:10:23,275 cd.
367 00:10:24,535 –> 00:10:26,151 Then it’s easy to see what
368 00:10:26,183 –> 00:10:27,751 the intersection is if you
369 00:10:27,783 –> 00:10:29,229 just do a short
370 00:10:29,277 –> 00:10:30,105 drawing.
371 00:10:31,365 –> 00:10:33,125 So here’s the first interval
372 00:10:33,285 –> 00:10:34,685 a to b.
373 00:10:34,845 –> 00:10:36,365 And maybe the second one
374 00:10:36,445 –> 00:10:37,613 is now here.
375 00:10:37,709 –> 00:10:39,677 So c to
376 00:10:39,741 –> 00:10:40,345 d.
377 00:10:41,005 –> 00:10:42,821 Hence if this is the positions
378 00:10:42,853 –> 00:10:44,309 of the intervals, then we
379 00:10:44,357 –> 00:10:46,157 immediately get out the empty
380 00:10:46,221 –> 00:10:46,825 set.
381 00:10:47,205 –> 00:10:48,341 And then you see, this is
382 00:10:48,373 –> 00:10:50,061 in the case when
383 00:10:50,253 –> 00:10:52,245 b is less or equal than
384 00:10:52,365 –> 00:10:53,077 c.
385 00:10:53,261 –> 00:10:54,865 Or the other way around,
386 00:10:55,195 –> 00:10:57,131 if d is less
387 00:10:57,203 –> 00:10:59,015 or equal than a.
388 00:10:59,595 –> 00:11:01,019 Of course this is not so
389 00:11:01,067 –> 00:11:01,659 exciting.
390 00:11:01,787 –> 00:11:03,283 So maybe more
391 00:11:03,339 –> 00:11:05,091 interesting would be if
392 00:11:05,123 –> 00:11:06,691 the interval is
393 00:11:06,843 –> 00:11:08,147 inside the other interval.
394 00:11:08,211 –> 00:11:10,139 So maybe we shift
395 00:11:10,187 –> 00:11:11,015 that around.
396 00:11:11,395 –> 00:11:13,307 So there we
397 00:11:13,331 –> 00:11:14,055 have it.
398 00:11:15,915 –> 00:11:17,851 Now what comes out here is
399 00:11:17,883 –> 00:11:19,571 the interval that starts
400 00:11:19,603 –> 00:11:21,109 with c and ends with
401 00:11:21,157 –> 00:11:21,745 b.
402 00:11:22,325 –> 00:11:23,837 So let’s put that in here.
403 00:11:23,901 –> 00:11:25,701 So I have c to
404 00:11:25,773 –> 00:11:27,581 b exactly in the
405 00:11:27,613 –> 00:11:29,509 case when my c
406 00:11:29,597 –> 00:11:31,277 lies inside
407 00:11:31,381 –> 00:11:32,705 interval ab,
408 00:11:33,565 –> 00:11:35,117 but d not.
409 00:11:35,301 –> 00:11:36,661 So I could write it in this
410 00:11:36,693 –> 00:11:37,265 way.
411 00:11:38,685 –> 00:11:39,917 Now you believe me that you
412 00:11:39,941 –> 00:11:41,821 get out the same or a similar
413 00:11:41,893 –> 00:11:42,917 result if you look at the
414 00:11:42,941 –> 00:11:44,221 symmetric case, so where
415 00:11:44,253 –> 00:11:45,613 the blue interval is on the
416 00:11:45,629 –> 00:11:46,749 right and the green intervals
417 00:11:46,797 –> 00:11:48,253 on the left, so no problem
418 00:11:48,309 –> 00:11:48,909 there.
419 00:11:49,077 –> 00:11:50,869 And that you also get such
420 00:11:50,917 –> 00:11:52,869 an interval out if the interval
421 00:11:52,917 –> 00:11:54,637 lies completely inside the
422 00:11:54,661 –> 00:11:55,465 other one.
423 00:11:55,885 –> 00:11:57,109 Therefore I don’t write this
424 00:11:57,157 –> 00:11:57,749 down.
425 00:11:57,917 –> 00:11:59,645 You note that we
426 00:11:59,765 –> 00:12:01,365 always get out
427 00:12:01,485 –> 00:12:02,549 such an interval.
428 00:12:02,637 –> 00:12:04,549 So an interval in A.
429 00:12:04,677 –> 00:12:06,077 So also this property
430 00:12:06,141 –> 00:12:07,845 holds and
431 00:12:07,925 –> 00:12:09,025 one holds.
432 00:12:09,365 –> 00:12:10,861 This means that we now have
433 00:12:10,893 –> 00:12:12,601 to check property three,
434 00:12:12,733 –> 00:12:14,285 the difference property.
435 00:12:14,825 –> 00:12:16,209 Here we look now at
436 00:12:16,297 –> 00:12:16,925 ab
437 00:12:18,145 –> 00:12:20,045 without the set
438 00:12:20,425 –> 00:12:22,005 cd.
439 00:12:22,785 –> 00:12:24,465 Again here let’s look at
440 00:12:24,545 –> 00:12:25,361 some cases.
441 00:12:25,473 –> 00:12:26,793 So maybe the first one from
442 00:12:26,849 –> 00:12:28,225 before, where there’s no
443 00:12:28,265 –> 00:12:29,049 intersection.
444 00:12:29,177 –> 00:12:30,449 Okay, then the difference
445 00:12:30,497 –> 00:12:30,905 is clear.
446 00:12:30,945 –> 00:12:32,605 We don’t subtract anything.
447 00:12:33,065 –> 00:12:34,857 The ab interval
448 00:12:34,961 –> 00:12:35,725 remains.
449 00:12:37,225 –> 00:12:38,787 Now in this case
450 00:12:38,921 –> 00:12:40,727 from here, you see, this
451 00:12:40,751 –> 00:12:41,839 is the intersection.
452 00:12:41,927 –> 00:12:43,591 So what remains is the
453 00:12:43,623 –> 00:12:45,559 interval from a to
454 00:12:45,607 –> 00:12:46,195 c.
455 00:12:47,215 –> 00:12:48,639 So let’s write this down
456 00:12:48,767 –> 00:12:50,687 a to c, but c
457 00:12:50,831 –> 00:12:52,515 is not included.
458 00:12:53,095 –> 00:12:54,831 Okay, maybe we can ignore
459 00:12:54,903 –> 00:12:56,367 the other symmetric case.
460 00:12:56,431 –> 00:12:57,927 So when the blue interval
461 00:12:57,951 –> 00:12:59,327 is on the right, then of
462 00:12:59,351 –> 00:13:00,855 course this interval would
463 00:13:00,895 –> 00:13:02,515 remain the right hand side.
464 00:13:03,575 –> 00:13:05,255 More interesting would be
465 00:13:05,295 –> 00:13:06,675 now another case
466 00:13:06,815 –> 00:13:08,379 where cd is
467 00:13:08,427 –> 00:13:09,695 inside ab.
468 00:13:10,395 –> 00:13:12,059 Of course for the intersection
469 00:13:12,107 –> 00:13:13,379 this was very boring.
470 00:13:13,507 –> 00:13:15,379 But now something happens
471 00:13:15,467 –> 00:13:16,095 here.
472 00:13:18,435 –> 00:13:20,187 Hence we subtract the middle
473 00:13:20,251 –> 00:13:21,171 part here.
474 00:13:21,323 –> 00:13:23,091 So what remains
475 00:13:23,243 –> 00:13:24,615 is this part
476 00:13:24,955 –> 00:13:26,815 and this part.
477 00:13:27,795 –> 00:13:29,659 In other words, this is a
478 00:13:29,707 –> 00:13:30,899 union of two
479 00:13:30,947 –> 00:13:32,743 intervals, namely
480 00:13:32,899 –> 00:13:34,355 a to c
481 00:13:34,895 –> 00:13:36,375 union with the
482 00:13:36,415 –> 00:13:38,007 interval d
483 00:13:38,151 –> 00:13:39,195 to b.
484 00:13:41,015 –> 00:13:42,783 But this is okay for a
485 00:13:42,799 –> 00:13:43,623 semi ring.
486 00:13:43,719 –> 00:13:45,511 We know we can have a union
487 00:13:45,583 –> 00:13:47,407 out of elements out
488 00:13:47,431 –> 00:13:48,275 of A.
489 00:13:49,175 –> 00:13:50,271 Okay, ignoring
490 00:13:50,343 –> 00:13:52,119 renaming of the letters
491 00:13:52,167 –> 00:13:53,967 here, we know these are all
492 00:13:53,991 –> 00:13:55,511 the cases that can happen
493 00:13:55,623 –> 00:13:57,555 and therefore we are safe.
494 00:13:58,305 –> 00:13:59,925 This is a semi ring.
495 00:14:00,865 –> 00:14:02,417 Okay, I hope this helped
496 00:14:02,441 –> 00:14:04,241 you a little bit about semi
497 00:14:04,273 –> 00:14:04,889 rings.
498 00:14:05,057 –> 00:14:06,889 And now we can talk about
499 00:14:06,977 –> 00:14:08,825 the next notion, which is
500 00:14:08,865 –> 00:14:10,085 the pre measure
501 00:14:11,185 –> 00:14:13,009 a premeasure is almost
502 00:14:13,097 –> 00:14:13,769 a measure.
503 00:14:13,857 –> 00:14:14,937 That’s the idea.
504 00:14:15,041 –> 00:14:16,961 But it’s not defined on a
505 00:14:16,993 –> 00:14:18,777 sigma algebra, but just
506 00:14:18,841 –> 00:14:20,393 on such a semi ring
507 00:14:20,449 –> 00:14:21,045 A.
508 00:14:21,825 –> 00:14:23,665 Ignoring this fact, the
509 00:14:23,705 –> 00:14:25,545 definition looks almost the
510 00:14:25,585 –> 00:14:27,549 same, which means we have
511 00:14:27,597 –> 00:14:28,557 two conditions.
512 00:14:28,701 –> 00:14:30,405 And the first one is that
513 00:14:30,445 –> 00:14:32,213 the pre measure of the empty
514 00:14:32,269 –> 00:14:34,077 set as
515 00:14:34,141 –> 00:14:35,029 the measure of the empty
516 00:14:35,077 –> 00:14:36,677 set should always be
517 00:14:36,741 –> 00:14:37,345 zero.
518 00:14:38,765 –> 00:14:40,365 And the second property
519 00:14:40,525 –> 00:14:41,877 was the sigma
520 00:14:41,941 –> 00:14:42,905 additivity.
521 00:14:43,685 –> 00:14:45,261 This is what we can write
522 00:14:45,413 –> 00:14:46,797 as mu
523 00:14:46,941 –> 00:14:48,665 of a union,
524 00:14:49,045 –> 00:14:50,661 a countable union.
525 00:14:50,773 –> 00:14:52,488 So J from 1
526 00:14:52,595 –> 00:14:54,099 to infinity of
527 00:14:54,207 –> 00:14:55,711 sets AJ is
528 00:14:55,818 –> 00:14:57,322 equal to a
529 00:14:57,429 –> 00:14:58,933 series which starts
530 00:14:59,041 –> 00:15:00,967 also with 1 and goes
531 00:15:00,991 –> 00:15:02,775 to infinity, where we have
532 00:15:02,815 –> 00:15:04,743 inside the three measures
533 00:15:04,879 –> 00:15:06,155 of AJ.
534 00:15:07,615 –> 00:15:09,431 Now what we need are
535 00:15:09,583 –> 00:15:11,143 sets AJ
536 00:15:11,319 –> 00:15:12,767 from our curved
537 00:15:12,831 –> 00:15:13,511 A.
538 00:15:13,703 –> 00:15:15,635 So from our semi ring.
539 00:15:15,935 –> 00:15:17,375 Of course, otherwise this
540 00:15:17,415 –> 00:15:18,675 does not make sense
541 00:15:19,345 –> 00:15:20,993 and there should be disjoint,
542 00:15:21,129 –> 00:15:22,073 which means
543 00:15:22,249 –> 00:15:24,185 AI intersected
544 00:15:24,225 –> 00:15:25,697 with aj
545 00:15:25,881 –> 00:15:27,793 is the empty set
546 00:15:27,969 –> 00:15:28,685 for
547 00:15:29,425 –> 00:15:30,913 I not equal
548 00:15:30,969 –> 00:15:31,565 j.
549 00:15:32,545 –> 00:15:34,273 Okay, so this was the definition
550 00:15:34,329 –> 00:15:36,121 or the sigma additivity for
551 00:15:36,153 –> 00:15:37,645 an ordinary measure.
552 00:15:38,185 –> 00:15:39,769 However, here it can’t be
553 00:15:39,817 –> 00:15:41,641 true, because we
554 00:15:41,673 –> 00:15:43,305 have here the union
555 00:15:43,465 –> 00:15:45,185 of sets out of
556 00:15:45,265 –> 00:15:45,901 A.
557 00:15:46,073 –> 00:15:47,853 For sigma algebra this is
558 00:15:47,869 –> 00:15:49,437 not a problem, because in
559 00:15:49,461 –> 00:15:50,549 the definition of a sigma
560 00:15:50,597 –> 00:15:52,341 algebra we already know
561 00:15:52,413 –> 00:15:54,301 this is also in A.
562 00:15:54,493 –> 00:15:56,381 However, for semi ring we
563 00:15:56,413 –> 00:15:57,301 don’t know this.
564 00:15:57,373 –> 00:15:59,357 We don’t know if this union
565 00:15:59,461 –> 00:16:01,305 is inside A as well.
566 00:16:01,805 –> 00:16:03,477 But we need this if we want
567 00:16:03,501 –> 00:16:05,421 to put this one into the
568 00:16:05,453 –> 00:16:07,197 map mu and
569 00:16:07,221 –> 00:16:09,005 therefore we have to add
570 00:16:09,165 –> 00:16:10,389 this condition here.
571 00:16:10,477 –> 00:16:12,175 So we add that
572 00:16:12,295 –> 00:16:13,687 the union here
573 00:16:13,871 –> 00:16:15,115 is also
574 00:16:15,935 –> 00:16:17,555 in A.
575 00:16:18,575 –> 00:16:19,847 This means you should read
576 00:16:19,911 –> 00:16:21,791 (b) in this way for all
577 00:16:21,863 –> 00:16:23,855 sets AJ where this
578 00:16:23,895 –> 00:16:25,159 one and this one is
579 00:16:25,207 –> 00:16:27,039 fulfilled, this one
580 00:16:27,087 –> 00:16:28,231 here is also
581 00:16:28,303 –> 00:16:29,195 satisfied.
582 00:16:29,855 –> 00:16:31,375 And then we can call the
583 00:16:31,415 –> 00:16:32,727 map mu a pre
584 00:16:32,751 –> 00:16:33,595 measure.
585 00:16:35,815 –> 00:16:37,255 Now, for the end of this
586 00:16:37,295 –> 00:16:39,049 video, I want to give you
587 00:16:39,137 –> 00:16:40,697 an important application
588 00:16:40,801 –> 00:16:42,409 of Carathéodory’s
589 00:16:42,457 –> 00:16:43,845 extension theorem.
590 00:16:44,385 –> 00:16:46,217 We have already seen
591 00:16:46,401 –> 00:16:47,697 that the finite
592 00:16:47,801 –> 00:16:49,361 intervals defined
593 00:16:49,473 –> 00:16:50,685 as ab
594 00:16:51,025 –> 00:16:52,177 where b is
595 00:16:52,241 –> 00:16:54,113 excluded form
596 00:16:54,169 –> 00:16:55,857 a semi ring of
597 00:16:55,921 –> 00:16:56,685 sets.
598 00:16:57,625 –> 00:16:59,513 Now we have no problem
599 00:16:59,689 –> 00:17:01,585 defining a pre measure,
600 00:17:01,705 –> 00:17:03,521 namely one that is defined
601 00:17:03,633 –> 00:17:05,473 on this finite
602 00:17:05,649 –> 00:17:06,525 intervals.
603 00:17:07,745 –> 00:17:09,665 What we can do is look at
604 00:17:09,705 –> 00:17:11,521 the length of the interval,
605 00:17:11,633 –> 00:17:13,289 which means our pre measure
606 00:17:13,337 –> 00:17:15,129 is defined as the
607 00:17:15,177 –> 00:17:16,993 difference between b
608 00:17:17,129 –> 00:17:18,085 and a.
609 00:17:18,785 –> 00:17:20,305 Now, it’s not hard to see
610 00:17:20,385 –> 00:17:22,137 that this is indeed a pre
611 00:17:22,161 –> 00:17:22,793 measure.
612 00:17:22,889 –> 00:17:24,497 By this definition, you can
613 00:17:24,521 –> 00:17:26,337 immediately check these two
614 00:17:26,401 –> 00:17:27,525 properties here.
615 00:17:28,745 –> 00:17:30,649 And now we can apply
616 00:17:30,777 –> 00:17:32,225 Carathéodory’s
617 00:17:32,265 –> 00:17:33,725 extension theorem
618 00:17:34,105 –> 00:17:35,393 and know that there is
619 00:17:35,449 –> 00:17:36,769 exactly one
620 00:17:36,857 –> 00:17:38,105 extension for this
621 00:17:38,145 –> 00:17:39,165 premeasure.
622 00:17:39,665 –> 00:17:41,225 And this measure now
623 00:17:41,265 –> 00:17:42,929 lives on the sigma
624 00:17:42,977 –> 00:17:44,417 algebra that is generated
625 00:17:44,481 –> 00:17:46,297 by A, so generated by
626 00:17:46,321 –> 00:17:48,121 the finite intervals here.
627 00:17:48,233 –> 00:17:49,673 And we already know the sigma
628 00:17:49,729 –> 00:17:50,601 algebra.
629 00:17:50,793 –> 00:17:52,145 It’s the Borel sigma
630 00:17:52,185 –> 00:17:52,965 algebra.
631 00:17:54,065 –> 00:17:55,689 Indeed, this is the measure
632 00:17:55,737 –> 00:17:57,561 that one knows as the
633 00:17:57,593 –> 00:17:58,765 Lebesgue measure.
634 00:17:59,355 –> 00:18:00,747 And here you see why
635 00:18:00,811 –> 00:18:02,139 Carathéodory’s
636 00:18:02,187 –> 00:18:03,643 theorem is so
637 00:18:03,699 –> 00:18:04,371 important.
638 00:18:04,563 –> 00:18:06,115 Because the Lebesgue measure
639 00:18:06,155 –> 00:18:07,135 is so important.
640 00:18:07,755 –> 00:18:09,579 And now by proving this
641 00:18:09,627 –> 00:18:11,419 extension theorem, we prove
642 00:18:11,467 –> 00:18:13,435 the existence and the uniqueness
643 00:18:13,515 –> 00:18:14,975 of the Lebesgue measure.
644 00:18:15,635 –> 00:18:17,203 Okay, there we have
645 00:18:17,219 –> 00:18:17,707 it.
646 00:18:17,851 –> 00:18:18,963 This was
647 00:18:18,979 –> 00:18:20,683 Carathéodory’s extension
648 00:18:20,779 –> 00:18:21,615 theorem.
649 00:18:22,075 –> 00:18:23,827 I really hope that this
650 00:18:23,891 –> 00:18:25,875 helped you a little bit understanding
651 00:18:25,915 –> 00:18:26,495 it.
652 00:18:26,945 –> 00:18:28,625 Of course, in other videos
653 00:18:28,665 –> 00:18:29,525 in future
654 00:18:29,825 –> 00:18:31,441 applications for this
655 00:18:31,473 –> 00:18:33,285 theorem will follow.
656 00:18:33,705 –> 00:18:35,673 And maybe we also talk about
657 00:18:35,729 –> 00:18:37,085 the proof later
658 00:18:38,025 –> 00:18:39,305 then thank you very much
659 00:18:39,345 –> 00:18:41,145 for listening and see you
660 00:18:41,185 –> 00:18:41,965 next time.
661 00:18:42,905 –> 00:18:43,225 Bye.
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