• Title: Fatou’s Lemma

  • Series: Measure Theory

  • YouTube-Title: Measure Theory 9 | Fatou’s Lemma

  • Bright video: https://youtu.be/qAYX9Koo87o

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  • Subtitle in English

    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.

  • Quiz Content

    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.

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