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Title: Outer measures - Part 1
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Series: Measure Theory
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YouTube-Title: Measure Theory 20 | Outer measures - Part 1
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Timestamps (n/a)
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Subtitle in English
1 00:00:00,709 –> 00:00:02,529 Hello and welcome back to
2 00:00:02,539 –> 00:00:04,210 new video about measure
3 00:00:04,219 –> 00:00:04,900 theory.
4 00:00:05,090 –> 00:00:06,369 And as always, I want to
5 00:00:06,380 –> 00:00:08,069 thank all the nice people
6 00:00:08,079 –> 00:00:09,270 that support this channel
7 00:00:09,279 –> 00:00:10,829 on Steady.
8 00:00:10,840 –> 00:00:11,520 In this,
9 00:00:11,529 –> 00:00:13,050 and in the next videos I
10 00:00:13,060 –> 00:00:14,840 want to talk about so called
11 00:00:14,850 –> 00:00:15,939 outer measures.
12 00:00:16,879 –> 00:00:18,430 And with this topic, we come
13 00:00:18,440 –> 00:00:20,030 back to the foundations of
14 00:00:20,040 –> 00:00:20,979 measure theory.
15 00:00:22,069 –> 00:00:23,709 In fact, after using Carathéodory
16 00:00:24,409 –> 00:00:26,159 extension theorem, many
17 00:00:26,170 –> 00:00:28,000 times, I now want to
18 00:00:28,010 –> 00:00:29,959 actually prove it. In
19 00:00:29,969 –> 00:00:30,950 order to do that,
20 00:00:30,959 –> 00:00:32,909 we need some suitable tools
21 00:00:33,020 –> 00:00:34,770 and we find out that the
22 00:00:34,779 –> 00:00:36,470 outer measures are indeed
23 00:00:36,479 –> 00:00:37,909 these important tools.
24 00:00:38,770 –> 00:00:40,450 For this reason, I can tell
25 00:00:40,459 –> 00:00:42,270 you that for understanding
26 00:00:42,279 –> 00:00:44,119 the idea of measure theory,
27 00:00:44,259 –> 00:00:45,490 you don’t really need the
28 00:00:45,500 –> 00:00:46,700 outer measures at all.
29 00:00:46,979 –> 00:00:48,310 But if you want to go into
30 00:00:48,319 –> 00:00:49,869 the technical details and
31 00:00:49,880 –> 00:00:51,389 maybe prove such an important
32 00:00:51,400 –> 00:00:53,090 theorem here, then you can’t
33 00:00:53,099 –> 00:00:54,380 avoid the outer measures.
34 00:00:55,439 –> 00:00:57,340 The name comes from the circumstance
35 00:00:57,349 –> 00:00:59,200 that one wants to approximate
36 00:00:59,209 –> 00:01:00,939 a set from the outside.
37 00:01:01,060 –> 00:01:02,220 But we will talk about this
38 00:01:02,229 –> 00:01:02,720 later.
39 00:01:03,340 –> 00:01:04,849 However, in this context,
40 00:01:04,860 –> 00:01:06,760 I should emphasize that both
41 00:01:06,769 –> 00:01:08,540 words here are connected.
42 00:01:08,569 –> 00:01:10,400 Outer measure is indeed a
43 00:01:10,410 –> 00:01:11,339 new notion.
44 00:01:12,290 –> 00:01:13,610 What I want to tell you here
45 00:01:13,620 –> 00:01:15,510 is that “outer” is
46 00:01:15,519 –> 00:01:16,989 not an attribute for
47 00:01:17,000 –> 00:01:18,540 measure. In
48 00:01:18,550 –> 00:01:19,260 particular,
49 00:01:19,269 –> 00:01:20,930 it’s important to note that
50 00:01:20,940 –> 00:01:22,660 outer measures don’t have
51 00:01:22,669 –> 00:01:24,480 to be measures in the
52 00:01:24,489 –> 00:01:25,220 usual sense.
53 00:01:26,190 –> 00:01:27,580 With this heads up at the
54 00:01:27,589 –> 00:01:29,319 beginning, I can however
55 00:01:29,330 –> 00:01:30,769 reassure you that the
56 00:01:30,779 –> 00:01:32,300 construction of such outer
57 00:01:32,309 –> 00:01:34,160 measures is not so complicated
58 00:01:34,169 –> 00:01:34,599 at all.
59 00:01:35,480 –> 00:01:37,099 Basically, I would say it’s
60 00:01:37,110 –> 00:01:38,680 even easier than for the
61 00:01:38,690 –> 00:01:40,580 usual measures we had before
62 00:01:40,750 –> 00:01:42,190 because we don’t need a Sigma
63 00:01:42,199 –> 00:01:43,220 algebra at the beginning.
64 00:01:44,050 –> 00:01:44,550 OK.
65 00:01:44,559 –> 00:01:45,730 I don’t want to keep you
66 00:01:45,739 –> 00:01:47,099 in suspense any longer,
67 00:01:47,300 –> 00:01:49,230 and therefore I present now
68 00:01:49,239 –> 00:01:50,150 the definition.
69 00:01:51,919 –> 00:01:53,260 Not so surprising,
70 00:01:53,269 –> 00:01:55,019 also an outer measure is
71 00:01:55,029 –> 00:01:56,629 first of all just a
72 00:01:56,639 –> 00:01:58,180 map defined
73 00:01:58,230 –> 00:01:59,819 on some subsets
74 00:01:59,830 –> 00:02:01,540 of a chosen fixed set
75 00:02:01,550 –> 00:02:02,019 X.
76 00:02:03,150 –> 00:02:04,970 Indeed admissible are now
77 00:02:04,980 –> 00:02:06,889 all subsets of X,
78 00:02:06,900 –> 00:02:08,169 and, therefore, we can choose
79 00:02:08,179 –> 00:02:08,970 the power set.
80 00:02:09,750 –> 00:02:11,008 And for the values we are
81 00:02:11,020 –> 00:02:12,970 already used to choose the
82 00:02:12,979 –> 00:02:14,449 non-negative numbers where
83 00:02:14,460 –> 00:02:15,690 we include the symbol
84 00:02:15,699 –> 00:02:16,529 infinity.
85 00:02:17,559 –> 00:02:19,460 Now, such a map which I
86 00:02:19,470 –> 00:02:21,220 now denote by Phi
87 00:02:21,229 –> 00:02:22,479 to distinguish it from a
88 00:02:22,490 –> 00:02:23,979 normal measure, which we
89 00:02:23,990 –> 00:02:25,619 would denote by mu is
90 00:02:25,630 –> 00:02:27,399 called an outer measure if
91 00:02:27,410 –> 00:02:29,050 the following three conditions
92 00:02:29,059 –> 00:02:29,860 are fulfilled.
93 00:02:30,630 –> 00:02:32,089 So please don’t forget that
94 00:02:32,100 –> 00:02:33,690 we always want to generalize
95 00:02:33,699 –> 00:02:35,449 or abstract how the
96 00:02:35,460 –> 00:02:37,259 normal measuring of volumes
97 00:02:37,270 –> 00:02:38,479 in space would work.
98 00:02:39,470 –> 00:02:40,649 Therefore, we would never
99 00:02:40,660 –> 00:02:42,160 demand any crazy
100 00:02:42,169 –> 00:02:43,729 properties from this function
101 00:02:43,740 –> 00:02:44,360 phi here.
102 00:02:45,270 –> 00:02:46,539 Hence the first property
103 00:02:46,550 –> 00:02:48,449 you already know if we put
104 00:02:48,460 –> 00:02:50,350 in the empty set in the map,
105 00:02:50,860 –> 00:02:52,809 we must get out zero.
106 00:02:54,139 –> 00:02:55,449 The empty set should not
107 00:02:55,460 –> 00:02:56,720 have any volume.
108 00:02:57,399 –> 00:02:59,110 Also the next property makes
109 00:02:59,119 –> 00:02:59,410 sense
110 00:02:59,419 –> 00:03:01,130 in this picture. If we look
111 00:03:01,139 –> 00:03:02,929 at a subset A of
112 00:03:02,940 –> 00:03:04,059 another set B
113 00:03:04,979 –> 00:03:06,570 and maybe we visualize that
114 00:03:06,580 –> 00:03:08,139 first here in the picture.
115 00:03:08,149 –> 00:03:09,690 So we have to set X, we have
116 00:03:09,699 –> 00:03:11,580 a subset B inside of the
117 00:03:11,589 –> 00:03:13,320 set and there are
118 00:03:13,330 –> 00:03:15,080 also a subset A
119 00:03:15,130 –> 00:03:16,399 inside of B.
120 00:03:18,059 –> 00:03:19,199 What does this mean for the
121 00:03:19,210 –> 00:03:20,020 volumes?
122 00:03:20,029 –> 00:03:21,639 Of course, the volume of
123 00:03:21,649 –> 00:03:23,559 A should be smaller or in
124 00:03:23,570 –> 00:03:25,139 the best case, the same as
125 00:03:25,149 –> 00:03:26,199 the volume of B.
126 00:03:27,240 –> 00:03:29,059 So we have here less or
127 00:03:29,070 –> 00:03:29,820 equal.
128 00:03:31,539 –> 00:03:32,820 And that is what we call
129 00:03:32,830 –> 00:03:34,820 the monotonicity of the outer
130 00:03:34,830 –> 00:03:35,279 measure.
131 00:03:36,130 –> 00:03:37,869 Please note this property
132 00:03:37,880 –> 00:03:39,869 we also had for an ordinary
133 00:03:39,880 –> 00:03:40,350 measure.
134 00:03:40,630 –> 00:03:42,070 It was just something we
135 00:03:42,080 –> 00:03:43,710 didn’t demand in the definition
136 00:03:43,830 –> 00:03:45,029 because it followed from
137 00:03:45,039 –> 00:03:45,990 the other properties.
138 00:03:47,270 –> 00:03:48,729 And now the third property
139 00:03:48,740 –> 00:03:50,320 is something you might also
140 00:03:50,330 –> 00:03:51,089 recognize.
141 00:03:52,009 –> 00:03:53,509 Let’s take a sequence of
142 00:03:53,520 –> 00:03:54,169 subsets,
143 00:03:54,179 –> 00:03:56,020 so let’s call them A
144 00:03:56,029 –> 00:03:57,630 one, A two and so on
145 00:03:57,679 –> 00:03:59,389 from the power set
146 00:03:59,399 –> 00:04:00,240 of X.
147 00:04:01,369 –> 00:04:02,639 And for this sequence then
148 00:04:02,649 –> 00:04:04,639 holds that the union
149 00:04:05,419 –> 00:04:06,850 measured with our map
150 00:04:06,860 –> 00:04:07,740 phi
151 00:04:08,759 –> 00:04:10,639 is smaller or equal
152 00:04:10,649 –> 00:04:12,419 than when we add up
153 00:04:12,429 –> 00:04:13,740 all the volumes
154 00:04:13,750 –> 00:04:14,649 separately.
155 00:04:15,699 –> 00:04:16,760 This is now something that
156 00:04:16,769 –> 00:04:18,238 we also can easily
157 00:04:18,250 –> 00:04:19,670 visualize in this picture
158 00:04:19,678 –> 00:04:20,040 here.
159 00:04:20,298 –> 00:04:22,239 So if we form the union
160 00:04:22,250 –> 00:04:23,829 of both A one and A
161 00:04:23,839 –> 00:04:25,350 two, well, we
162 00:04:25,359 –> 00:04:27,320 calculate the area here
163 00:04:27,540 –> 00:04:28,440 which is of course
164 00:04:28,450 –> 00:04:30,019 smaller than
165 00:04:30,029 –> 00:04:31,640 adding the volume or the
166 00:04:31,649 –> 00:04:33,380 area here for A one
167 00:04:33,429 –> 00:04:35,230 plus the area of A
168 00:04:35,239 –> 00:04:37,000 two simply because
169 00:04:37,010 –> 00:04:38,399 we add this intersection
170 00:04:38,410 –> 00:04:39,359 two times.
171 00:04:40,239 –> 00:04:41,760 You see, we don’t need
172 00:04:41,769 –> 00:04:43,380 disjoint sets here, which means
173 00:04:43,390 –> 00:04:45,079 we can have intersections
174 00:04:45,089 –> 00:04:46,519 and they occur on the right
175 00:04:46,529 –> 00:04:47,179 hand side.
176 00:04:47,260 –> 00:04:48,730 So in general, we will
177 00:04:48,739 –> 00:04:50,390 have an inequality in
178 00:04:50,399 –> 00:04:51,140 general here.
179 00:04:51,970 –> 00:04:53,200 And because we have it for
180 00:04:53,209 –> 00:04:54,959 countable many subsets,
181 00:04:55,190 –> 00:04:56,440 you see, we have an index
182 00:04:56,450 –> 00:04:57,619 in the natural numbers N.
183 00:04:57,630 –> 00:04:59,489 here, we call it the
184 00:04:59,500 –> 00:05:00,709 sigma-subadditivity.
185 00:05:01,869 –> 00:05:03,690 And that is all we need to
186 00:05:03,700 –> 00:05:04,970 get an outer measure.
187 00:05:05,730 –> 00:05:07,329 When you see these properties,
188 00:05:07,339 –> 00:05:09,040 you might think this is
189 00:05:09,049 –> 00:05:10,779 indeed a good substitution
190 00:05:10,790 –> 00:05:12,739 or abstraction for
191 00:05:12,750 –> 00:05:14,559 measuring normal volumes.
192 00:05:15,339 –> 00:05:17,049 However, you already know
193 00:05:17,079 –> 00:05:18,790 that this can’t work for
194 00:05:18,799 –> 00:05:20,450 the whole power set if you
195 00:05:20,459 –> 00:05:22,000 consider the important Lebesgue
196 00:05:22,010 –> 00:05:22,450 measure.
197 00:05:23,239 –> 00:05:24,529 And you also see here in
198 00:05:24,540 –> 00:05:26,519 part C that we don’t have
199 00:05:26,529 –> 00:05:28,519 the equality if we consider
200 00:05:28,529 –> 00:05:30,049 pairwise disjoint sets
201 00:05:30,059 –> 00:05:31,739 something we usually call
202 00:05:31,750 –> 00:05:33,079 Sigma additivity.
203 00:05:34,109 –> 00:05:35,489 Hence a natural question
204 00:05:35,500 –> 00:05:37,390 here is if I start
205 00:05:37,399 –> 00:05:39,010 with an outer measure, Phi,
206 00:05:39,070 –> 00:05:40,549 how do I get an ordinary
207 00:05:40,559 –> 00:05:41,529 measure mu?
208 00:05:42,070 –> 00:05:43,480 Or in other words, how do
209 00:05:43,489 –> 00:05:45,279 I get also for the map
210 00:05:45,290 –> 00:05:47,179 Phi the Sigma additivity?
211 00:05:48,149 –> 00:05:49,809 And we already know this
212 00:05:49,820 –> 00:05:51,040 won’t work for the whole
213 00:05:51,049 –> 00:05:51,709 power set.
214 00:05:52,359 –> 00:05:53,700 Therefore, the first thing
215 00:05:53,709 –> 00:05:55,000 we should do is to
216 00:05:55,010 –> 00:05:56,579 restrict the sets we
217 00:05:56,589 –> 00:05:57,220 consider.
218 00:05:57,929 –> 00:05:59,549 And this leads us to our
219 00:05:59,559 –> 00:06:00,670 next definition,
220 00:06:01,640 –> 00:06:03,220 we only want to look at some
221 00:06:03,230 –> 00:06:05,019 good sets that we
222 00:06:05,029 –> 00:06:05,820 can measure.
223 00:06:06,019 –> 00:06:07,059 And in the end, they will
224 00:06:07,070 –> 00:06:08,500 form a Sigma algebra again.
225 00:06:09,579 –> 00:06:11,260 And the name for such a
226 00:06:11,269 –> 00:06:12,369 good set
227 00:06:12,690 –> 00:06:13,760 is phi-
228 00:06:14,010 –> 00:06:15,019 measurable.
229 00:06:16,010 –> 00:06:17,429 And we use that name
230 00:06:17,760 –> 00:06:19,260 if for all other
231 00:06:19,269 –> 00:06:20,040 subsets.
232 00:06:20,649 –> 00:06:21,820 And now I choose to letter
233 00:06:21,829 –> 00:06:23,579 Q for another subset,
234 00:06:24,679 –> 00:06:26,390 we have the following add
235 00:06:26,399 –> 00:06:27,390 activity rule.
236 00:06:28,059 –> 00:06:29,809 The best thing is again to
237 00:06:29,820 –> 00:06:31,440 visualize that in a picture.
238 00:06:31,450 –> 00:06:33,160 So there’s a set A and
239 00:06:33,170 –> 00:06:35,000 here’s another set Q.
240 00:06:36,230 –> 00:06:38,019 Now the set A splits
241 00:06:38,029 –> 00:06:39,959 this set Q into two
242 00:06:39,970 –> 00:06:41,170 parts, the one
243 00:06:41,179 –> 00:06:43,089 inside A and the
244 00:06:43,100 –> 00:06:44,850 other one outside
245 00:06:44,859 –> 00:06:45,209 A.
246 00:06:46,390 –> 00:06:48,170 Or by using formulas, here
247 00:06:48,179 –> 00:06:49,720 we have the inner part and
248 00:06:49,730 –> 00:06:51,130 here we have the part outside
249 00:06:51,140 –> 00:06:52,910 A and now we
250 00:06:52,920 –> 00:06:54,600 measure both sets by
251 00:06:54,609 –> 00:06:56,119 using our map phi
252 00:06:56,440 –> 00:06:57,459 and add them up.
253 00:06:58,929 –> 00:07:00,390 And now we would say this
254 00:07:00,399 –> 00:07:02,029 measurement is meaningful
255 00:07:02,040 –> 00:07:03,869 if we get out the measure
256 00:07:03,880 –> 00:07:04,339 of Q.
257 00:07:04,350 –> 00:07:05,920 Again, this
258 00:07:05,929 –> 00:07:07,519 means that we call a set
259 00:07:07,529 –> 00:07:08,799 a phi-
260 00:07:08,809 –> 00:07:10,579 measurable if it’s so
261 00:07:10,589 –> 00:07:12,140 good that it divides
262 00:07:12,149 –> 00:07:13,920 every other set Q
263 00:07:13,929 –> 00:07:15,739 into two parts such
264 00:07:15,750 –> 00:07:17,670 that there our map
265 00:07:17,679 –> 00:07:19,559 phi is indeed additive.
266 00:07:20,730 –> 00:07:22,559 Obviously, this is the first
267 00:07:22,570 –> 00:07:24,350 step to the Sigma additivity
268 00:07:24,359 –> 00:07:24,920 we want.
269 00:07:25,839 –> 00:07:26,899 Now, what I really should
270 00:07:26,910 –> 00:07:28,579 add here is that in some
271 00:07:28,589 –> 00:07:30,059 books, you might find the
272 00:07:30,070 –> 00:07:31,559 definition as
273 00:07:31,649 –> 00:07:33,100 greater and equal.
274 00:07:33,880 –> 00:07:35,260 The reason for that is that
275 00:07:35,269 –> 00:07:36,649 it does not make any difference
276 00:07:36,660 –> 00:07:38,260 at all because for an outer
277 00:07:38,269 –> 00:07:39,579 measure, you have the other
278 00:07:39,589 –> 00:07:41,179 inequality here in the
279 00:07:41,190 –> 00:07:41,820 definition.
280 00:07:42,730 –> 00:07:44,290 Hence, you could define it
281 00:07:44,299 –> 00:07:45,549 in this weaker sense.
282 00:07:45,559 –> 00:07:47,190 But still, it means the same
283 00:07:47,200 –> 00:07:47,470 thing.
284 00:07:48,279 –> 00:07:49,899 Well, let’s end this video
285 00:07:49,910 –> 00:07:51,140 with some cliffhanger.
286 00:07:51,170 –> 00:07:52,350 But of course, I will go
287 00:07:52,359 –> 00:07:53,980 into more detail in the next
288 00:07:53,989 –> 00:07:54,459 video.
289 00:07:55,290 –> 00:07:56,589 The important proposition
290 00:07:56,600 –> 00:07:58,290 here is written in the following
291 00:07:58,299 –> 00:08:00,290 way, if we have an
292 00:08:00,299 –> 00:08:01,989 outer measure on the
293 00:08:02,000 –> 00:08:02,980 set X,
294 00:08:03,660 –> 00:08:05,640 then the collection of all
295 00:08:05,649 –> 00:08:07,359 these good sets here.
296 00:08:07,420 –> 00:08:09,160 And let’s call them A
297 00:08:09,170 –> 00:08:10,529 with index Phi
298 00:08:11,429 –> 00:08:13,260 forms a Sigma
299 00:08:13,269 –> 00:08:13,929 algebra.
300 00:08:14,959 –> 00:08:16,390 Of course, that makes our
301 00:08:16,399 –> 00:08:18,109 good sets even better
302 00:08:18,500 –> 00:08:19,970 because this is the first
303 00:08:19,980 –> 00:08:21,630 part we need for our measure.
304 00:08:22,640 –> 00:08:24,470 In fact, if we define a map
305 00:08:24,480 –> 00:08:26,399 mu on the
306 00:08:26,410 –> 00:08:28,089 Sigma algebra A phi
307 00:08:29,440 –> 00:08:30,739 and how can we define it
308 00:08:30,750 –> 00:08:31,739 very easily?
309 00:08:31,750 –> 00:08:33,030 We just use our Phi
310 00:08:33,039 –> 00:08:34,820 from before.
311 00:08:34,830 –> 00:08:36,580 So we have mu
312 00:08:36,590 –> 00:08:38,229 of A is the same
313 00:08:38,239 –> 00:08:40,099 thing as putting A into
314 00:08:40,109 –> 00:08:41,869 phi and what
315 00:08:41,880 –> 00:08:43,679 comes out is a beautiful
316 00:08:43,690 –> 00:08:44,320 measure.
317 00:08:45,419 –> 00:08:46,929 So you see this is
318 00:08:46,940 –> 00:08:48,450 indeed a very nice
319 00:08:48,460 –> 00:08:49,109 result.
320 00:08:49,909 –> 00:08:51,169 And then in the next video,
321 00:08:51,219 –> 00:08:53,090 I will show you some examples
322 00:08:53,099 –> 00:08:54,530 of outer measures and then
323 00:08:54,539 –> 00:08:55,900 you will see that it’s not
324 00:08:55,909 –> 00:08:57,489 hard at all to construct
325 00:08:57,500 –> 00:08:57,729 them.
326 00:08:58,739 –> 00:09:00,309 In addition, this proposition
327 00:09:00,320 –> 00:09:01,909 then will help us to get
328 00:09:01,919 –> 00:09:03,750 out an ordinary measure as
329 00:09:03,760 –> 00:09:04,309 we want.
330 00:09:05,140 –> 00:09:06,169 And of course, if you’re
331 00:09:06,179 –> 00:09:07,500 interested, we will also
332 00:09:07,510 –> 00:09:08,799 prove this proposition.
333 00:09:09,640 –> 00:09:10,179 OK.
334 00:09:10,190 –> 00:09:11,520 Thank you for listening and
335 00:09:11,530 –> 00:09:12,500 see you next time.
336 00:09:12,520 –> 00:09:13,140 Bye.
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Last update: 2024-10
