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 x-axis 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 x-axis 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 y-axis, 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 x-axis

148 00:04:08,440 –> 00:04:09,839 and the set on the x-axis

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.

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.