1 00:00:00,639 –> 00:00:02,509 Hello and welcome back to

2 00:00:02,519 –> 00:00:03,750 measure theory.

3 00:00:03,759 –> 00:00:04,230 First.

4 00:00:04,239 –> 00:00:05,619 Let me thank all the nice

5 00:00:05,630 –> 00:00:07,300 supporters on Steady.

6 00:00:08,500 –> 00:00:09,880 We have come very far in

7 00:00:09,890 –> 00:00:11,510 a series and reached now

8 00:00:11,520 –> 00:00:12,949 part nine,

9 00:00:14,090 –> 00:00:15,600 I was talking about the very

10 00:00:15,609 –> 00:00:17,530 important convergence theorems.

11 00:00:17,540 –> 00:00:19,090 And today I want to tell

12 00:00:19,100 –> 00:00:20,549 you about Fatou’s Lemma.

13 00:00:21,659 –> 00:00:23,440 This lemma is much more

14 00:00:23,450 –> 00:00:24,690 important than it sounds

15 00:00:24,700 –> 00:00:25,450 first.

16 00:00:25,469 –> 00:00:26,959 However, it’s not

17 00:00:26,969 –> 00:00:28,700 quite a convergence theorem,

18 00:00:28,709 –> 00:00:30,559 more or less half a convergence

19 00:00:30,569 –> 00:00:32,400 theorem, but it is

20 00:00:32,409 –> 00:00:34,220 so general that you can apply

21 00:00:34,229 –> 00:00:35,400 it very often.

22 00:00:36,080 –> 00:00:36,500 OK.

23 00:00:36,509 –> 00:00:38,439 Then let me explain you

24 00:00:38,450 –> 00:00:40,020 what the Lemma states.

25 00:00:40,830 –> 00:00:42,580 As always, we have a measure

26 00:00:42,590 –> 00:00:44,220 space consisting

27 00:00:44,229 –> 00:00:45,900 of a set, a Sigma

28 00:00:45,909 –> 00:00:47,099 algebra and a measure.

29 00:00:47,810 –> 00:00:49,409 And we also have a

30 00:00:49,419 –> 00:00:51,349 sequence of measurable

31 00:00:51,360 –> 00:00:52,189 functions.

32 00:00:53,060 –> 00:00:53,639 OK.

33 00:00:53,650 –> 00:00:54,810 Here I want to write down

34 00:00:54,819 –> 00:00:56,500 the codomain of the functions

35 00:00:56,689 –> 00:00:57,759 and there I have to tell

36 00:00:57,770 –> 00:00:58,759 you there are different

37 00:00:58,770 –> 00:01:00,639 possibilities and they

38 00:01:00,650 –> 00:01:02,549 lead to different variations

39 00:01:02,560 –> 00:01:03,509 of Fatou’s lemma.

40 00:01:04,330 –> 00:01:05,949 However, I think the best

41 00:01:05,959 –> 00:01:07,910 way to describe it is again

42 00:01:07,919 –> 00:01:09,360 to choose non-negative

43 00:01:09,790 –> 00:01:10,580 functions.

44 00:01:11,089 –> 00:01:12,900 When I say non negative,

45 00:01:12,910 –> 00:01:14,349 then it’s clear that the

46 00:01:14,360 –> 00:01:15,970 zero is included, but it’s

47 00:01:15,980 –> 00:01:17,900 not clear if infinity

48 00:01:17,910 –> 00:01:18,580 is included.

49 00:01:19,370 –> 00:01:20,309 However, here, that’s not

50 00:01:20,319 –> 00:01:20,720 a problem.

51 00:01:20,730 –> 00:01:22,110 We can just include it.

52 00:01:23,790 –> 00:01:25,449 And now to the statement

53 00:01:25,459 –> 00:01:26,650 of this lemma,

54 00:01:27,379 –> 00:01:28,680 because we are in the realm

55 00:01:28,690 –> 00:01:30,360 of the convergence theorems,

56 00:01:30,370 –> 00:01:32,120 the question would be again,

57 00:01:32,269 –> 00:01:34,050 OK, I have integral over

58 00:01:34,059 –> 00:01:35,319 X and a limit

59 00:01:35,330 –> 00:01:36,279 inside.

60 00:01:36,290 –> 00:01:37,879 Can I pull out

61 00:01:37,889 –> 00:01:38,980 this limit?

62 00:01:41,269 –> 00:01:42,589 However, Fatou’s Lemma

63 00:01:42,599 –> 00:01:44,269 is a little bit more specific,

64 00:01:44,279 –> 00:01:46,150 it does not look at the limit

65 00:01:46,489 –> 00:01:48,319 rather than at the limit

66 00:01:48,330 –> 00:01:49,199 inferior.

67 00:01:50,010 –> 00:01:51,750 Now, Fatou’s Lemma claims

68 00:01:51,849 –> 00:01:53,489 that if we look at the limit

69 00:01:53,500 –> 00:01:55,239 inferior of our sequence

70 00:01:55,250 –> 00:01:57,050 of functions FN, so

71 00:01:57,059 –> 00:01:57,790 here’s d mu

72 00:02:00,750 –> 00:02:02,230 then I can pull out the limit

73 00:02:02,239 –> 00:02:03,809 inferior and then I have

74 00:02:03,819 –> 00:02:05,809 to limit inferior here of

75 00:02:05,819 –> 00:02:07,430 the numbers given by the

76 00:02:07,440 –> 00:02:08,770 integrals of FN.

77 00:02:09,600 –> 00:02:11,220 But we don’t have an

78 00:02:11,229 –> 00:02:12,679 equality sign here,

79 00:02:12,990 –> 00:02:14,750 just an inequality sign.

80 00:02:15,089 –> 00:02:16,690 And Fatou tells us the

81 00:02:16,699 –> 00:02:18,429 left hand side can’t be

82 00:02:18,440 –> 00:02:19,809 bigger than the right hand

83 00:02:19,820 –> 00:02:20,350 side.

84 00:02:21,649 –> 00:02:23,369 So you see the Lemma is not

85 00:02:23,380 –> 00:02:24,789 so strong as a convergence

86 00:02:24,800 –> 00:02:25,389 theorem.

87 00:02:25,619 –> 00:02:27,160 But please note that our

88 00:02:27,169 –> 00:02:29,059 requirements are very weak.

89 00:02:29,509 –> 00:02:31,059 We only need non-negative

90 00:02:31,550 –> 00:02:33,190 measurable maps.

91 00:02:33,199 –> 00:02:34,059 Nothing else.

92 00:02:34,750 –> 00:02:36,449 The actual convergence

93 00:02:36,460 –> 00:02:37,759 theorem that follows from

94 00:02:37,770 –> 00:02:39,350 this claim here is Lebesgue’s

95 00:02:39,720 –> 00:02:41,410 theorem which we will consider

96 00:02:41,419 –> 00:02:42,199 in the next video

97 00:02:43,149 –> 00:02:44,389 before showing you the proof

98 00:02:44,399 –> 00:02:45,460 of this nice lemma.

99 00:02:45,479 –> 00:02:47,320 Let me first explain

100 00:02:47,330 –> 00:02:49,210 what this new function liminf

101 00:02:49,220 –> 00:02:51,089 in of the sequence of functions

102 00:02:51,100 –> 00:02:52,169 actually is.

103 00:02:52,979 –> 00:02:54,139 Well, of course, it is a

104 00:02:54,149 –> 00:02:54,960 function

105 00:02:56,339 –> 00:02:57,600 which means it is

106 00:02:57,610 –> 00:02:59,460 defined on X

107 00:02:59,470 –> 00:03:00,919 and maps also

108 00:03:01,029 –> 00:03:02,710 into our non negative

109 00:03:02,720 –> 00:03:04,039 numbers including

110 00:03:04,440 –> 00:03:05,360 infinity.

111 00:03:06,429 –> 00:03:07,539 And now I want to give you

112 00:03:07,550 –> 00:03:08,449 the definition.

113 00:03:08,460 –> 00:03:09,770 So the functions are defined

114 00:03:09,779 –> 00:03:11,490 for all lower case X.

115 00:03:12,509 –> 00:03:14,089 And now we use what we know

116 00:03:14,100 –> 00:03:15,679 from the limit inferior.

117 00:03:15,880 –> 00:03:17,309 It is given by the

118 00:03:17,320 –> 00:03:18,940 limit of the infima

119 00:03:20,929 –> 00:03:22,809 but cut it at the beginning.

120 00:03:22,820 –> 00:03:24,029 So we go from

121 00:03:24,839 –> 00:03:26,339 N to infinity

122 00:03:27,360 –> 00:03:28,929 and then we look what happens

123 00:03:28,990 –> 00:03:30,779 when n goes to infinity.

124 00:03:32,089 –> 00:03:33,210 This is the definition of

125 00:03:33,220 –> 00:03:34,779 the limit inferior for a

126 00:03:34,789 –> 00:03:36,070 sequence of numbers.

127 00:03:36,080 –> 00:03:37,490 And here we just have numbers

128 00:03:37,500 –> 00:03:38,759 because we look at

129 00:03:39,350 –> 00:03:39,750 FX

130 00:03:41,460 –> 00:03:43,039 and they will see the lowest

131 00:03:43,050 –> 00:03:44,440 level that the infimum can

132 00:03:44,449 –> 00:03:46,160 have would be zero.

133 00:03:46,729 –> 00:03:48,520 But in the limit, of

134 00:03:48,529 –> 00:03:50,279 course, it could go to infinity.

135 00:03:50,289 –> 00:03:52,220 So we also have to include

136 00:03:52,229 –> 00:03:53,119 infinity

137 00:03:53,889 –> 00:03:55,259 also in the case when we

138 00:03:55,270 –> 00:03:56,960 would exclude infinity

139 00:03:56,970 –> 00:03:58,690 here because in a limit it

140 00:03:58,699 –> 00:03:59,380 could happen.

141 00:03:59,880 –> 00:04:01,520 And because we need the infinity

142 00:04:01,529 –> 00:04:03,100 symbol for this function,

143 00:04:03,110 –> 00:04:04,910 we can just add it from the

144 00:04:04,919 –> 00:04:05,550 beginning.

145 00:04:06,270 –> 00:04:07,770 And of course, it just makes

146 00:04:07,779 –> 00:04:08,889 the claim stronger.

147 00:04:09,839 –> 00:04:11,300 The beauty of this is that

148 00:04:11,309 –> 00:04:12,600 we know that the limit

149 00:04:12,610 –> 00:04:14,070 inferior is also

150 00:04:14,080 –> 00:04:14,970 measurable

151 00:04:15,639 –> 00:04:17,209 simply because you can easily

152 00:04:17,220 –> 00:04:18,959 show that if you put in

153 00:04:18,970 –> 00:04:20,940 measurable functions in

154 00:04:21,000 –> 00:04:22,600 infima are measurable

155 00:04:23,019 –> 00:04:24,779 and also limits of

156 00:04:24,790 –> 00:04:26,179 measurable functions are

157 00:04:26,190 –> 00:04:27,649 also measurable

158 00:04:28,160 –> 00:04:29,059 for the whole proof.

159 00:04:29,070 –> 00:04:30,899 Now it makes sense to use

160 00:04:30,910 –> 00:04:32,579 some abbreviations here,

161 00:04:33,510 –> 00:04:35,010 let’s call the limit inferior

162 00:04:35,019 –> 00:04:36,820 just by G of X.

163 00:04:36,829 –> 00:04:38,010 So we have the function G

164 00:04:38,019 –> 00:04:39,290 now and

165 00:04:39,299 –> 00:04:41,130 also these functions

166 00:04:41,140 –> 00:04:42,839 here given by the

167 00:04:42,850 –> 00:04:44,790 infima let’s call them

168 00:04:44,799 –> 00:04:46,529 GN of X.

169 00:04:47,399 –> 00:04:49,350 And of course, all the functions

170 00:04:49,359 –> 00:04:50,899 here are measurable

171 00:04:51,670 –> 00:04:53,130 and we get another information

172 00:04:53,140 –> 00:04:53,730 out here.

173 00:04:53,739 –> 00:04:55,609 These functions here are

174 00:04:55,619 –> 00:04:57,570 now monotonically increasing.

175 00:04:58,029 –> 00:04:59,720 So G one is less or

176 00:04:59,730 –> 00:05:01,190 equal than G two,

177 00:05:01,290 –> 00:05:02,690 less or equal than G

178 00:05:02,700 –> 00:05:03,390 three.

179 00:05:03,619 –> 00:05:04,850 And so on,

180 00:05:05,859 –> 00:05:07,130 this follows immediately

181 00:05:07,140 –> 00:05:08,779 from the definition of the

182 00:05:08,790 –> 00:05:09,480 infimum.

183 00:05:10,170 –> 00:05:11,799 Because if we shift the cut

184 00:05:11,809 –> 00:05:13,510 point and to divide

185 00:05:13,779 –> 00:05:15,709 the inimum can only get bigger,

186 00:05:15,720 –> 00:05:16,839 not smaller.

187 00:05:17,600 –> 00:05:17,989 OK.

188 00:05:18,000 –> 00:05:19,829 So we get out a sequence

189 00:05:19,839 –> 00:05:21,109 that is monotonically

190 00:05:21,119 –> 00:05:21,910 increasing.

191 00:05:22,609 –> 00:05:24,029 And of course, this would

192 00:05:24,040 –> 00:05:25,690 be very helpful for our proof

193 00:05:25,700 –> 00:05:27,489 now because we can

194 00:05:27,500 –> 00:05:29,250 use our convergence theorem,

195 00:05:29,380 –> 00:05:30,390 we already know.

196 00:05:30,750 –> 00:05:31,790 And of course, you know,

197 00:05:31,799 –> 00:05:33,589 it’s the monotone convergence

198 00:05:33,600 –> 00:05:34,109 theorem.

199 00:05:35,100 –> 00:05:36,279 And indeed, with this, the

200 00:05:36,290 –> 00:05:37,820 proof is not so long

201 00:05:38,880 –> 00:05:40,100 on the left hand side, we

202 00:05:40,109 –> 00:05:41,540 have the limit inferior.

203 00:05:41,549 –> 00:05:43,459 And now I write that as

204 00:05:43,470 –> 00:05:45,209 the limit of our

205 00:05:45,220 –> 00:05:45,989 GNS

206 00:05:47,130 –> 00:05:48,859 indeed, that is just our

207 00:05:48,869 –> 00:05:49,980 limit inferior here.

208 00:05:51,089 –> 00:05:52,619 And now we want to pull out

209 00:05:52,630 –> 00:05:53,549 the limit here.

210 00:05:54,869 –> 00:05:56,570 And you know, now it’s allowed

211 00:05:56,579 –> 00:05:58,230 by the monotone convergence

212 00:05:58,619 –> 00:05:58,640 theorem.

213 00:05:59,839 –> 00:06:01,730 And we also have the equality

214 00:06:01,739 –> 00:06:02,100 here.

215 00:06:02,850 –> 00:06:04,260 Please check with the last

216 00:06:04,269 –> 00:06:06,109 videos that we have indeed

217 00:06:06,119 –> 00:06:08,079 satisfied all the requirements

218 00:06:08,089 –> 00:06:08,579 of the theorem.

219 00:06:10,000 –> 00:06:11,380 Now the next step, I

220 00:06:11,390 –> 00:06:13,140 substitute the limit

221 00:06:13,149 –> 00:06:14,660 with the limit inferior

222 00:06:16,100 –> 00:06:17,529 simply because this is the

223 00:06:17,540 –> 00:06:18,760 thing we want to talk about

224 00:06:18,769 –> 00:06:19,559 in Fatou’s Lemma.

225 00:06:21,000 –> 00:06:22,510 And of course here it’s the

226 00:06:22,519 –> 00:06:24,119 same thing doesn’t make any

227 00:06:24,130 –> 00:06:25,140 difference at all.

228 00:06:26,190 –> 00:06:28,000 However, in the end, we want

229 00:06:28,010 –> 00:06:29,739 FN and not GN in the

230 00:06:29,750 –> 00:06:30,609 integral here.

231 00:06:31,489 –> 00:06:33,010 Therefore, you can ask, what

232 00:06:33,019 –> 00:06:34,179 is the connection between

233 00:06:34,190 –> 00:06:35,709 G and and FN

234 00:06:36,570 –> 00:06:37,410 and then you look at the

235 00:06:37,420 –> 00:06:38,829 definition and see,

236 00:06:39,019 –> 00:06:40,640 OK, the G and is defined

237 00:06:40,649 –> 00:06:42,429 about all FK where

238 00:06:42,440 –> 00:06:43,869 K is bigger than N

239 00:06:44,609 –> 00:06:46,250 and then you choose the infimum,

240 00:06:46,260 –> 00:06:48,029 the smallest possible value.

241 00:06:48,679 –> 00:06:50,209 And therefore, of course

242 00:06:50,220 –> 00:06:52,179 GN is always less or

243 00:06:52,190 –> 00:06:53,790 equal than FN as

244 00:06:53,799 –> 00:06:54,640 being the infimum.

245 00:06:56,019 –> 00:06:57,769 So let’s say that as we

246 00:06:57,779 –> 00:06:58,290 know

247 00:06:59,040 –> 00:07:00,920 GN, less or equal to FN

248 00:07:00,929 –> 00:07:02,640 for all N and

249 00:07:02,649 –> 00:07:03,950 now you want to use the

250 00:07:03,959 –> 00:07:05,679 monotonicity of the integral,

251 00:07:05,980 –> 00:07:07,570 which is a very nice property

252 00:07:07,579 –> 00:07:08,809 of the Lebesgue integral.

253 00:07:08,820 –> 00:07:10,630 So if we have the inequality

254 00:07:10,640 –> 00:07:12,149 here, then it also

255 00:07:12,160 –> 00:07:14,049 holds for the integral.

256 00:07:16,179 –> 00:07:16,799 OK.

257 00:07:16,809 –> 00:07:18,760 So this inequality we now

258 00:07:18,769 –> 00:07:20,589 want to use here on the

259 00:07:20,600 –> 00:07:21,519 right hand side.

260 00:07:23,420 –> 00:07:24,649 So let me summarize what

261 00:07:24,660 –> 00:07:25,410 we have here.

262 00:07:26,190 –> 00:07:27,570 The left hand side is the

263 00:07:27,579 –> 00:07:28,890 limit inferior

264 00:07:28,899 –> 00:07:30,690 inside the integral.

265 00:07:32,179 –> 00:07:33,760 And on the right hand side,

266 00:07:33,769 –> 00:07:34,899 we can use this

267 00:07:34,910 –> 00:07:36,420 inequality, the limit

268 00:07:36,429 –> 00:07:38,040 inferior, conserve the

269 00:07:38,049 –> 00:07:39,940 inequality, which

270 00:07:39,950 –> 00:07:41,910 means we have here less or

271 00:07:41,920 –> 00:07:43,239 equal and limit

272 00:07:43,250 –> 00:07:44,899 inferior outside

273 00:07:47,670 –> 00:07:49,329 FNDmu.

274 00:07:50,720 –> 00:07:52,299 And there we have it that

275 00:07:52,309 –> 00:07:53,619 is Fatou’s Lemma.

276 00:07:56,119 –> 00:07:57,720 You see the proof was not

277 00:07:57,730 –> 00:07:59,619 so hard and not so long because

278 00:07:59,630 –> 00:08:01,480 we could use our monotone

279 00:08:01,489 –> 00:08:02,859 convergence theorem here.

280 00:08:04,010 –> 00:08:04,549 OK.

281 00:08:04,559 –> 00:08:06,260 So I hope you learned something

282 00:08:06,269 –> 00:08:07,940 today and then you will see

283 00:08:07,950 –> 00:08:09,660 in the next video how we

284 00:08:09,670 –> 00:08:11,540 can apply Fatou’s lemma.

285 00:08:12,420 –> 00:08:14,359 And then we can finally prove

286 00:08:14,369 –> 00:08:15,880 one of my favorite

00:08:15,890 –> 00:08:16,760 theorems.

288 00:08:17,500 –> 00:08:18,160 OK.

289 00:08:18,170 –> 00:08:19,779 I wish you a very nice day

290 00:08:19,790 –> 00:08:21,239 and see you next time.

291 00:08:21,899 –> 00:08:22,359 Bye.

Q1: Let $(X, \mathcal{A}, \mu)$ be a measure space and $f_n: X \rightarrow [0,\infty]$ be measurable for all $n \in \mathbb{N}$. What is the correct formulation for Fatou’s lemma?

A1: $$ \int_X f_n , d\mu = \int_X \lim_{n\rightarrow \infty} f_n , d\mu$$

A2: $$ \int_X f_n , d\mu \leq \int_X \liminf_{n\rightarrow \infty} f_n , d\mu$$

A3: $$ \int_X \liminf_{n\rightarrow \infty} f_n , d\mu \leq \liminf_{n\rightarrow \infty} \int_X f_n , d\mu$$

A4: $$ \int_X \liminf_{n\rightarrow \infty} f_n , d\mu \geq \liminf_{n\rightarrow \infty} \int_X f_n , d\mu$$

Q2: Did we use the monotone convergence theorem in the proof of Fatou’s Lemma?

A1: Yes

A2: No

Q3: Let $(X, \mathcal{A}, \mu)$ be a measure space and $f_n: X \rightarrow [0,\infty]$ be measurable for all $n \in \mathbb{N}$. Is it possible have $$ \int_X \liminf_{n\rightarrow \infty} f_n , d\mu = \liminf_{n\rightarrow \infty} \int_X f_n , d\mu$$

A1: Yes, there are examples.

A2: No, Fatou’s Lemma does not allow it.