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.