• Title: Monotone Convergence Theorem (Proof and Application)

  • Series: Measure Theory

  • YouTube-Title: Measure Theory 8 | Monotone Convergence Theorem (Proof and Application)

  • Bright video: https://youtu.be/iUXQBxwyr-o

  • Dark video: https://youtu.be/_Qb_qVjV2bU

  • Quiz: Test your knowledge

  • PDF: Download PDF version of the bright video

  • Dark-PDF: Download PDF version of the dark video

  • Print-PDF: Download printable PDF version

  • Thumbnail (bright): Download PNG

  • Thumbnail (dark): Download PNG

  • Subtitle on GitHub: mt08_sub_eng.srt

  • Other languages: German version

  • Timestamps (n/a)
  • Subtitle in English

    1 00:00:00,670 –> 00:00:02,529 Hello and welcome back to

    2 00:00:02,539 –> 00:00:03,970 the next video in this

    3 00:00:03,980 –> 00:00:04,769 series.

    4 00:00:05,280 –> 00:00:07,269 I thank all the nice people

    5 00:00:07,280 –> 00:00:08,689 that support this channel

    6 00:00:08,699 –> 00:00:09,630 on Steady

    7 00:00:10,510 –> 00:00:12,470 Well, this is measure theory

    8 00:00:12,479 –> 00:00:13,779 part eight.

    9 00:00:13,930 –> 00:00:15,359 And we are still talking

    10 00:00:15,369 –> 00:00:17,049 about the monotone convergence

    11 00:00:17,059 –> 00:00:17,670 theorem.

    12 00:00:17,809 –> 00:00:19,590 But today I want to show

    13 00:00:19,600 –> 00:00:21,329 you mainly the proof

    14 00:00:21,340 –> 00:00:22,840 of this theorem

    15 00:00:24,010 –> 00:00:25,600 for the start of this video.

    16 00:00:25,709 –> 00:00:27,540 It might be helpful to

    17 00:00:27,549 –> 00:00:29,350 give you a recap what the

    18 00:00:29,360 –> 00:00:30,899 monotone convergence theorem

    19 00:00:30,909 –> 00:00:32,729 states. For the

    20 00:00:32,740 –> 00:00:34,479 ingredients, we need a measure

    21 00:00:34,490 –> 00:00:35,939 space and

    22 00:00:35,950 –> 00:00:37,380 measurable maps

    23 00:00:38,139 –> 00:00:39,549 as always they are non-

    24 00:00:39,560 –> 00:00:40,200 negative.

    25 00:00:40,389 –> 00:00:41,880 And we have a whole sequence

    26 00:00:41,889 –> 00:00:43,520 of such measurable maps.

    27 00:00:44,040 –> 00:00:45,720 And in addition, we also

    28 00:00:45,729 –> 00:00:47,169 have the limit where we call

    29 00:00:47,180 –> 00:00:48,720 this just F without an

    30 00:00:48,729 –> 00:00:49,279 index.

    31 00:00:50,049 –> 00:00:50,389 OK.

    32 00:00:50,400 –> 00:00:52,119 Because the theorem is called

    33 00:00:52,130 –> 00:00:53,869 monotone convergence theorem.

    34 00:00:53,880 –> 00:00:55,669 We need two things first,

    35 00:00:55,680 –> 00:00:57,349 a monotonic property and

    36 00:00:57,360 –> 00:00:58,869 also a convergent property.

    37 00:00:59,680 –> 00:01:01,340 The first thing is therefore

    38 00:01:01,349 –> 00:01:02,509 the sequence is

    39 00:01:02,520 –> 00:01:04,379 monotonically increasing

    40 00:01:05,069 –> 00:01:07,049 but only almost everywhere

    41 00:01:07,058 –> 00:01:08,579 with respect to our measure

    42 00:01:08,588 –> 00:01:08,869 mu

    43 00:01:09,558 –> 00:01:11,049 And the second property I

    44 00:01:11,058 –> 00:01:12,888 already told you F should

    45 00:01:12,899 –> 00:01:14,329 be the limit of

    46 00:01:14,338 –> 00:01:16,308 FN and the limit should

    47 00:01:16,319 –> 00:01:18,219 be in the pointwise sense which

    48 00:01:18,228 –> 00:01:19,719 means I fix a point

    49 00:01:19,728 –> 00:01:21,369 X and then I have a normal

    50 00:01:21,378 –> 00:01:23,109 limit of real numbers.

    51 00:01:23,359 –> 00:01:25,088 However, as before, this

    52 00:01:25,098 –> 00:01:26,738 property only has to hold

    53 00:01:26,749 –> 00:01:28,088 almost everywhere.

    54 00:01:28,900 –> 00:01:30,639 So much about the premise.

    55 00:01:30,650 –> 00:01:32,400 Now let’s talk about the

    56 00:01:32,410 –> 00:01:33,169 conclusion.

    57 00:01:33,830 –> 00:01:35,610 The theorem tells us that

    58 00:01:35,620 –> 00:01:37,069 we can push the

    59 00:01:37,080 –> 00:01:38,790 limit into the

    60 00:01:38,800 –> 00:01:39,599 integral.

    61 00:01:39,720 –> 00:01:41,410 This means that the sequence

    62 00:01:41,419 –> 00:01:43,150 of integrals converges

    63 00:01:43,160 –> 00:01:45,150 to an integral where in the

    64 00:01:45,160 –> 00:01:46,779 integral there’s the

    65 00:01:46,790 –> 00:01:48,650 limit of the sequence.

    66 00:01:51,000 –> 00:01:52,989 And of course, this is just

    67 00:01:53,000 –> 00:01:54,050 our F here.

    68 00:01:55,910 –> 00:01:57,550 And now we have again our

    69 00:01:57,559 –> 00:01:59,050 monotone convergence

    70 00:01:59,059 –> 00:01:59,569 theorem.

    71 00:02:00,260 –> 00:02:02,050 However, I should tell you

    72 00:02:02,059 –> 00:02:04,050 there is an alternative formulation

    73 00:02:04,059 –> 00:02:05,750 of this theorem which is

    74 00:02:05,760 –> 00:02:07,250 a little bit more general

    75 00:02:08,100 –> 00:02:09,679 what you do there is you

    76 00:02:09,690 –> 00:02:10,750 include also

    77 00:02:10,758 –> 00:02:12,619 functions that have the

    78 00:02:12,630 –> 00:02:14,479 symbol infinity as an

    79 00:02:14,490 –> 00:02:15,160 outcome.

    80 00:02:15,919 –> 00:02:17,580 Of course, then you have

    81 00:02:17,589 –> 00:02:19,270 to extend the Borel Sigma

    82 00:02:19,279 –> 00:02:20,860 algebra to this

    83 00:02:20,869 –> 00:02:22,380 new bigger set

    84 00:02:22,710 –> 00:02:24,229 and also extend the notion

    85 00:02:24,240 –> 00:02:25,199 of measurable.

    86 00:02:25,750 –> 00:02:26,809 And indeed, that’s not a

    87 00:02:26,820 –> 00:02:27,740 problem at all

    88 00:02:29,070 –> 00:02:30,000 in the same way.

    89 00:02:30,009 –> 00:02:31,509 Also the definition of the

    90 00:02:31,520 –> 00:02:32,740 Integral does not change

    91 00:02:32,750 –> 00:02:34,309 at all because in the worst

    92 00:02:34,320 –> 00:02:35,919 case, as always, it could

    93 00:02:35,929 –> 00:02:36,839 be infinity.

    94 00:02:37,720 –> 00:02:39,279 The advantage of this formulation

    95 00:02:39,289 –> 00:02:40,990 is now that you can get rid

    96 00:02:41,000 –> 00:02:42,470 of the F completely.

    97 00:02:43,250 –> 00:02:45,009 So we don’t need this

    98 00:02:45,020 –> 00:02:46,660 and we don’t need

    99 00:02:46,669 –> 00:02:47,470 this

    100 00:02:48,479 –> 00:02:50,179 simply because everything

    101 00:02:50,229 –> 00:02:52,009 follows from the monotonic

    102 00:02:52,020 –> 00:02:53,800 property here, we have

    103 00:02:53,809 –> 00:02:55,619 measurable functions and

    104 00:02:55,630 –> 00:02:57,130 we know the supremum is also

    105 00:02:57,139 –> 00:02:58,949 measurable and they are

    106 00:02:58,960 –> 00:03:00,619 monotonically increasing.

    107 00:03:00,669 –> 00:03:02,419 So we know the supremum is given

    108 00:03:02,429 –> 00:03:03,789 almost everywhere as the

    109 00:03:03,800 –> 00:03:04,259 limit.

    110 00:03:05,350 –> 00:03:06,369 And there you see this is

    111 00:03:06,380 –> 00:03:08,369 just a more compact formulation

    112 00:03:08,380 –> 00:03:09,470 of the theorem.

    113 00:03:09,770 –> 00:03:11,320 And as I told you before,

    114 00:03:11,330 –> 00:03:12,380 in the worst case, you have

    115 00:03:12,389 –> 00:03:13,979 infinity left and

    116 00:03:13,990 –> 00:03:15,729 right in the conclusion here.

    117 00:03:17,880 –> 00:03:19,419 Well, so this was the

    118 00:03:19,429 –> 00:03:20,300 introduction.

    119 00:03:20,309 –> 00:03:22,229 Now, let’s look at

    120 00:03:22,240 –> 00:03:23,000 the proof,

    121 00:03:25,929 –> 00:03:27,220 of course, we use what we

    122 00:03:27,229 –> 00:03:28,199 know we have a

    123 00:03:28,210 –> 00:03:29,389 monotonically

    124 00:03:29,399 –> 00:03:31,309 increasing sequence

    125 00:03:31,320 –> 00:03:32,389 of functions.

    126 00:03:34,270 –> 00:03:36,009 This property holds almost

    127 00:03:36,020 –> 00:03:36,759 everywhere.

    128 00:03:37,050 –> 00:03:38,330 And if you remember what

    129 00:03:38,339 –> 00:03:39,710 we did in the last video,

    130 00:03:39,740 –> 00:03:41,039 we know that we have the

    131 00:03:41,050 –> 00:03:42,880 monotonicity property of

    132 00:03:42,889 –> 00:03:43,970 the Lebesgue integral.

    133 00:03:44,470 –> 00:03:46,089 This means that the integral

    134 00:03:46,100 –> 00:03:47,669 conserves

    135 00:03:47,679 –> 00:03:49,429 these inequalities.

    136 00:03:52,839 –> 00:03:54,320 Now we have a monotonically

    137 00:03:54,330 –> 00:03:56,210 increasing sequence of real

    138 00:03:56,220 –> 00:03:57,600 numbers where also

    139 00:03:57,610 –> 00:03:59,059 infinity can occur

    140 00:03:59,990 –> 00:04:01,440 in the same sense, we can

    141 00:04:01,449 –> 00:04:03,250 write down the inequality

    142 00:04:03,259 –> 00:04:04,270 for all N.

    143 00:04:04,279 –> 00:04:05,929 So we know that the limit

    144 00:04:06,179 –> 00:04:08,009 is always bigger or equal

    145 00:04:08,020 –> 00:04:09,850 than FN for

    146 00:04:09,860 –> 00:04:11,639 all N and almost

    147 00:04:11,649 –> 00:04:12,429 everywhere,

    148 00:04:13,619 –> 00:04:14,800 simply because the limit

    149 00:04:14,809 –> 00:04:16,279 is almost everywhere, the

    150 00:04:16,290 –> 00:04:18,200 supremum and because

    151 00:04:18,209 –> 00:04:19,920 this holds almost everywhere,

    152 00:04:19,928 –> 00:04:21,829 we know it holds for the

    153 00:04:21,850 –> 00:04:23,309 integral as well.

    154 00:04:24,269 –> 00:04:26,179 And there you see, we already

    155 00:04:26,190 –> 00:04:27,690 have one part of the

    156 00:04:27,700 –> 00:04:28,720 equality here.

    157 00:04:28,730 –> 00:04:29,649 We want to show.

    158 00:04:29,799 –> 00:04:31,489 So let’s call this star

    159 00:04:31,500 –> 00:04:32,850 for a reference

    160 00:04:33,989 –> 00:04:35,630 and let’s show the one

    161 00:04:35,640 –> 00:04:36,380 inequality.

    162 00:04:36,390 –> 00:04:37,970 So we formed the limit on

    163 00:04:37,980 –> 00:04:39,309 both sides here.

    164 00:04:39,929 –> 00:04:41,070 Of course, we don’t change

    165 00:04:41,079 –> 00:04:42,160 the inequality here.

    166 00:04:42,170 –> 00:04:43,609 And we have no limit on the

    167 00:04:43,619 –> 00:04:44,809 right hand side because there’s

    168 00:04:44,820 –> 00:04:46,429 no N involved.

    169 00:04:46,440 –> 00:04:48,170 And therefore we have this

    170 00:04:48,179 –> 00:04:49,329 as the inequality.

    171 00:04:50,049 –> 00:04:51,510 And this is exactly one

    172 00:04:51,519 –> 00:04:53,420 part of our

    173 00:04:53,429 –> 00:04:54,739 equality star.

    174 00:04:55,850 –> 00:04:57,269 Do you see this was not so

    175 00:04:57,279 –> 00:04:57,790 hard.

    176 00:04:57,799 –> 00:04:59,579 This part comes immediately

    177 00:05:00,380 –> 00:05:01,959 but for the other part we

    178 00:05:01,970 –> 00:05:03,410 need to use the definition

    179 00:05:03,420 –> 00:05:04,640 of the Lebesgue integral.

    180 00:05:05,179 –> 00:05:06,839 And this always means that

    181 00:05:06,850 –> 00:05:08,790 we go back to step or simple

    182 00:05:08,799 –> 00:05:09,709 functions.

    183 00:05:09,829 –> 00:05:11,480 Therefore, let h be an

    184 00:05:11,489 –> 00:05:13,390 arbitrary chosen simple function

    185 00:05:13,529 –> 00:05:15,130 that lies below

    186 00:05:15,140 –> 00:05:16,559 our function F

    187 00:05:19,089 –> 00:05:20,239 and also choose a

    188 00:05:20,250 –> 00:05:21,709 fixed error

    189 00:05:21,720 –> 00:05:22,589 epsilon

    190 00:05:23,809 –> 00:05:24,559 at this point.

    191 00:05:24,570 –> 00:05:26,049 I think a short sketch is

    192 00:05:26,059 –> 00:05:27,730 very helpful to get the idea.

    193 00:05:28,880 –> 00:05:30,720 Here is our function F

    194 00:05:31,140 –> 00:05:32,779 and this one should be the

    195 00:05:32,790 –> 00:05:34,630 step function H so maybe

    196 00:05:34,640 –> 00:05:35,559 it looks like this.

    197 00:05:37,890 –> 00:05:38,299 OK.

    198 00:05:38,309 –> 00:05:40,220 So this would be the usual

    199 00:05:40,230 –> 00:05:40,799 idea.

    200 00:05:41,720 –> 00:05:43,609 Now we know we have the pointwise

    201 00:05:43,619 –> 00:05:45,160 convergence of this

    202 00:05:45,170 –> 00:05:46,500 sequence of functions to

    203 00:05:46,510 –> 00:05:47,160 F.

    204 00:05:47,170 –> 00:05:48,850 So maybe we can choose some

    205 00:05:48,859 –> 00:05:50,179 FN that lies

    206 00:05:51,019 –> 00:05:52,600 very close to F.

    207 00:05:52,709 –> 00:05:54,359 So maybe this is

    208 00:05:54,369 –> 00:05:56,029 our chosen FN.

    209 00:05:57,299 –> 00:05:59,040 The idea of the epsilon is

    210 00:05:59,049 –> 00:06:00,929 now that we can always push

    211 00:06:00,940 –> 00:06:02,709 down the step function by

    212 00:06:02,720 –> 00:06:03,640 this epsilon.

    213 00:06:03,869 –> 00:06:05,390 So this would be epsilon

    214 00:06:05,399 –> 00:06:06,619 and then the new step function

    215 00:06:06,630 –> 00:06:07,549 looks like this.

    216 00:06:08,220 –> 00:06:09,859 And this should then always

    217 00:06:09,869 –> 00:06:11,760 be a step function that lies

    218 00:06:11,769 –> 00:06:13,000 below FN.

    219 00:06:14,200 –> 00:06:15,859 Hence I define now

    220 00:06:15,869 –> 00:06:17,739 sets XN

    221 00:06:17,929 –> 00:06:19,660 where I put in all the

    222 00:06:19,670 –> 00:06:21,109 points X

    223 00:06:22,290 –> 00:06:24,089 where FN of X

    224 00:06:24,200 –> 00:06:25,549 is bigger than the

    225 00:06:25,559 –> 00:06:27,260 shifted step function.

    226 00:06:27,269 –> 00:06:28,609 And the shifted step function

    227 00:06:28,619 –> 00:06:30,190 is just one minus

    228 00:06:30,200 –> 00:06:32,029 epsilon h

    229 00:06:34,200 –> 00:06:35,299 because we have our

    230 00:06:35,309 –> 00:06:36,700 convergence of the

    231 00:06:36,709 –> 00:06:38,670 sequence FN to F

    232 00:06:38,679 –> 00:06:39,899 almost everywhere

    233 00:06:40,420 –> 00:06:41,859 we know that

    234 00:06:42,089 –> 00:06:43,899 almost every X

    235 00:06:43,980 –> 00:06:45,730 in X lies

    236 00:06:45,739 –> 00:06:47,410 in at least one of these

    237 00:06:47,420 –> 00:06:47,929 XN.

    238 00:06:48,929 –> 00:06:50,480 In other words, we can look

    239 00:06:50,489 –> 00:06:52,399 at the union of all these

    240 00:06:52,410 –> 00:06:54,369 XN and

    241 00:06:54,380 –> 00:06:56,369 let’s call it X tilde,

    242 00:06:56,850 –> 00:06:58,450 then we know that the complement

    243 00:06:58,459 –> 00:07:00,450 of X tilde is a set with

    244 00:07:00,459 –> 00:07:01,489 measure zero.

    245 00:07:04,390 –> 00:07:05,950 Knowing this, we can now

    246 00:07:05,959 –> 00:07:07,429 look at the integral again.

    247 00:07:08,549 –> 00:07:10,010 Now we start here with the

    248 00:07:10,019 –> 00:07:11,089 left hand side.

    249 00:07:11,100 –> 00:07:12,670 So I want to look at the

    250 00:07:12,679 –> 00:07:14,529 integral of FN.

    251 00:07:16,309 –> 00:07:17,890 This integral gets not

    252 00:07:17,899 –> 00:07:19,850 bigger if I change the set

    253 00:07:19,859 –> 00:07:20,850 to a smaller set.

    254 00:07:20,859 –> 00:07:22,160 So here I want to choose

    255 00:07:22,170 –> 00:07:23,010 XN

    256 00:07:24,260 –> 00:07:25,700 Fn dmu.

    257 00:07:26,350 –> 00:07:28,329 Now inside the set XN,

    258 00:07:28,339 –> 00:07:30,239 we have this inequality

    259 00:07:30,299 –> 00:07:31,570 and can use the normal

    260 00:07:31,579 –> 00:07:33,390 monotonicity of the integral.

    261 00:07:34,260 –> 00:07:35,250 Therefore, I have the same

    262 00:07:35,260 –> 00:07:36,500 inequality here.

    263 00:07:36,510 –> 00:07:38,109 And a new function inside

    264 00:07:38,119 –> 00:07:39,480 the integral which is now

    265 00:07:39,489 –> 00:07:40,959 one minus epsilon

    266 00:07:40,970 –> 00:07:42,480 h dmu.

    267 00:07:43,750 –> 00:07:45,070 And of course, we want to

    268 00:07:45,079 –> 00:07:46,209 look at the limit.

    269 00:07:46,369 –> 00:07:48,089 So we found the limit here

    270 00:07:48,100 –> 00:07:49,250 on both sides,

    271 00:07:49,880 –> 00:07:50,859 then we get

    272 00:07:51,519 –> 00:07:52,420 limit

    273 00:07:53,399 –> 00:07:54,829 of the integrals.

    274 00:07:55,709 –> 00:07:57,250 And on the right hand side,

    275 00:07:57,290 –> 00:07:59,089 we have the limit of

    276 00:07:59,100 –> 00:08:00,329 these integrals.

    277 00:08:01,149 –> 00:08:01,769 OK.

    278 00:08:01,970 –> 00:08:03,720 At this point, I want to

    279 00:08:03,730 –> 00:08:05,529 omit some details because

    280 00:08:05,540 –> 00:08:06,869 otherwise we will lose our

    281 00:08:06,880 –> 00:08:07,989 focus completely.

    282 00:08:08,380 –> 00:08:10,010 What one can show now is

    283 00:08:10,019 –> 00:08:11,890 that this is indeed

    284 00:08:12,250 –> 00:08:13,869 the integral over

    285 00:08:13,950 –> 00:08:14,720 X tilde.

    286 00:08:15,260 –> 00:08:17,170 And we don’t change the

    287 00:08:17,179 –> 00:08:18,320 function at all.

    288 00:08:19,459 –> 00:08:20,690 And what we need in the details

    289 00:08:20,700 –> 00:08:22,239 is of course that this is

    290 00:08:22,250 –> 00:08:23,109 a step function.

    291 00:08:23,119 –> 00:08:24,820 So the integral is very easy.

    292 00:08:25,140 –> 00:08:26,510 Essentially, it’s just a

    293 00:08:26,519 –> 00:08:27,880 sum of our measures.

    294 00:08:28,619 –> 00:08:30,420 And we know that the

    295 00:08:30,429 –> 00:08:31,799 sets here are

    296 00:08:31,809 –> 00:08:33,760 increasing, you

    297 00:08:33,770 –> 00:08:35,619 have to find the sets appropriately.

    298 00:08:35,630 –> 00:08:36,830 But then you get out that

    299 00:08:36,840 –> 00:08:38,260 you have a monotonically

    300 00:08:38,270 –> 00:08:40,229 increasing sequence

    301 00:08:40,239 –> 00:08:41,020 of sets.

    302 00:08:41,789 –> 00:08:42,450 OK.

    303 00:08:42,640 –> 00:08:43,700 In summary, I’ll leave the

    304 00:08:43,710 –> 00:08:44,969 details here for you.

    305 00:08:44,979 –> 00:08:46,950 But remember we can do

    306 00:08:46,960 –> 00:08:48,580 this because this is just

    307 00:08:48,590 –> 00:08:50,010 a step function and integral.

    308 00:08:50,020 –> 00:08:51,010 It’s very easy there.

    309 00:08:51,840 –> 00:08:52,200 OK.

    310 00:08:52,210 –> 00:08:54,099 But after this, you see it

    311 00:08:54,109 –> 00:08:55,219 doesn’t make any difference

    312 00:08:55,229 –> 00:08:57,059 to choose X tilde here or X

    313 00:08:58,280 –> 00:08:59,359 because for the integral,

    314 00:08:59,369 –> 00:09:01,320 it does not matter what happens

    315 00:09:01,409 –> 00:09:03,140 on a set with measure

    316 00:09:03,150 –> 00:09:03,729 zero.

    317 00:09:04,280 –> 00:09:05,299 And in the last video, I

    318 00:09:05,309 –> 00:09:06,799 have shown it explicitly

    319 00:09:06,809 –> 00:09:07,969 for the step functions.

    320 00:09:09,229 –> 00:09:09,929 OK.

    321 00:09:09,940 –> 00:09:10,849 Now we have an

    322 00:09:10,859 –> 00:09:12,729 inequality that holds

    323 00:09:12,739 –> 00:09:14,530 for all epsilon here on the

    324 00:09:14,539 –> 00:09:16,409 right hand side, this

    325 00:09:16,419 –> 00:09:18,219 means now that we can choose

    326 00:09:18,229 –> 00:09:19,630 the epsilon as

    327 00:09:19,640 –> 00:09:21,340 small as we want.

    328 00:09:22,229 –> 00:09:23,900 And therefore we have a

    329 00:09:23,909 –> 00:09:25,719 limit process for the epsilon

    330 00:09:25,729 –> 00:09:26,989 and still have the

    331 00:09:27,000 –> 00:09:28,650 inequality if we

    332 00:09:28,659 –> 00:09:30,530 would set epsilon to

    333 00:09:30,539 –> 00:09:31,289 zero.

    334 00:09:31,299 –> 00:09:33,159 So this means here we

    335 00:09:33,169 –> 00:09:34,750 have the integral greater

    336 00:09:34,760 –> 00:09:36,390 or equal than the

    337 00:09:36,400 –> 00:09:37,229 integral.

    338 00:09:37,309 –> 00:09:38,469 And here just h

    339 00:09:38,909 –> 00:09:39,909 remains.

    340 00:09:41,609 –> 00:09:43,270 Now in the same sense,

    341 00:09:43,280 –> 00:09:44,909 this h was arbitrarily

    342 00:09:44,919 –> 00:09:45,479 chosen.

    343 00:09:45,989 –> 00:09:47,669 It was just a step function

    344 00:09:47,679 –> 00:09:49,229 that lies below our

    345 00:09:49,239 –> 00:09:50,299 function F.

    346 00:09:51,440 –> 00:09:52,940 This means now that we can

    347 00:09:52,950 –> 00:09:53,909 go to the supremum

    348 00:09:54,219 –> 00:09:56,049 over all the step

    349 00:09:56,059 –> 00:09:56,880 functions.

    350 00:09:57,020 –> 00:09:58,229 And then we get on the right

    351 00:09:58,239 –> 00:09:59,940 hand side, the integral of

    352 00:09:59,950 –> 00:10:01,710 F and there you have

    353 00:10:01,719 –> 00:10:01,960 it.

    354 00:10:01,969 –> 00:10:03,630 This is the second part of

    355 00:10:03,640 –> 00:10:05,260 our equality star.

    356 00:10:05,710 –> 00:10:07,070 Both things together

    357 00:10:07,210 –> 00:10:08,960 gives us the proof of the

    358 00:10:08,969 –> 00:10:10,789 monotone convergence theorem.

    359 00:10:11,940 –> 00:10:12,330 OK.

    360 00:10:12,340 –> 00:10:13,719 For the end of the video,

    361 00:10:13,849 –> 00:10:15,450 I want to show you an

    362 00:10:15,460 –> 00:10:17,090 application of this very

    363 00:10:17,099 –> 00:10:18,869 important monotone convergence

    364 00:10:18,880 –> 00:10:19,320 theorem.

    365 00:10:20,510 –> 00:10:22,409 One can always use this

    366 00:10:22,419 –> 00:10:24,409 for series if

    367 00:10:24,419 –> 00:10:26,309 you have any sequence of

    368 00:10:26,320 –> 00:10:27,960 functions, then I use now the

    369 00:10:27,979 –> 00:10:29,799 letter G that are

    370 00:10:29,809 –> 00:10:31,479 non-negative and also

    371 00:10:31,489 –> 00:10:32,369 measurable.

    372 00:10:33,440 –> 00:10:35,119 Then you can apply the

    373 00:10:35,130 –> 00:10:36,900 monotone convergence theorem.

    374 00:10:37,530 –> 00:10:39,219 So here I use also to assume

    375 00:10:39,229 –> 00:10:41,030 the infinity for the functions

    376 00:10:41,289 –> 00:10:43,179 and measurable.

    377 00:10:44,030 –> 00:10:45,659 So we don’t need a monotonic

    378 00:10:45,669 –> 00:10:46,580 behavior here.

    379 00:10:47,890 –> 00:10:49,770 Then we can say that the

    380 00:10:49,780 –> 00:10:50,710 series

    381 00:10:51,320 –> 00:10:53,239 which now goes from one

    382 00:10:53,250 –> 00:10:54,140 to infinity.

    383 00:10:54,820 –> 00:10:56,309 This is also a well defined

    384 00:10:56,320 –> 00:10:58,010 function and also measurable.

    385 00:10:59,099 –> 00:11:00,659 The trick is now of course,

    386 00:11:00,669 –> 00:11:02,239 that the partial sums

    387 00:11:02,250 –> 00:11:03,700 form a monotonically

    388 00:11:03,710 –> 00:11:05,179 increasing sequence of

    389 00:11:05,190 –> 00:11:05,840 functions.

    390 00:11:07,119 –> 00:11:08,840 Therefore, we can write down

    391 00:11:08,849 –> 00:11:10,539 the integral of the

    392 00:11:10,549 –> 00:11:11,419 series.

    393 00:11:11,510 –> 00:11:13,179 So it goes to infinity.

    394 00:11:13,559 –> 00:11:15,539 g n de mu

    395 00:11:16,320 –> 00:11:18,179 is equal, putting

    396 00:11:18,190 –> 00:11:20,080 the series outside the

    397 00:11:20,090 –> 00:11:20,599 integrals

    398 00:11:21,349 –> 00:11:23,340 So here’s the series of

    399 00:11:23,349 –> 00:11:24,429 the integrals

    400 00:11:24,710 –> 00:11:25,869 So gn

    401 00:11:26,059 –> 00:11:26,770 d mu

    402 00:11:27,780 –> 00:11:29,729 this means that the monotone

    403 00:11:29,739 –> 00:11:31,690 convergence theorem allows

    404 00:11:31,700 –> 00:11:33,130 us to switch

    405 00:11:33,140 –> 00:11:34,599 integral and series.

    406 00:11:35,479 –> 00:11:36,520 The only thing we have to

    407 00:11:36,530 –> 00:11:38,299 put in is that we have

    408 00:11:38,309 –> 00:11:40,039 non-negative measurable

    409 00:11:40,049 –> 00:11:41,880 functions and we

    410 00:11:41,890 –> 00:11:43,469 need no other assumptions

    411 00:11:43,479 –> 00:11:43,960 at all.

    412 00:11:44,789 –> 00:11:46,049 And there you see the power

    413 00:11:46,059 –> 00:11:47,710 of this theorem, it gives

    414 00:11:47,719 –> 00:11:49,179 you immediately such a nice

    415 00:11:49,190 –> 00:11:50,250 property as this

    416 00:11:51,450 –> 00:11:52,309 very well.

    417 00:11:52,500 –> 00:11:53,919 That’s all I wanted to tell

    418 00:11:53,929 –> 00:11:55,830 you about the monotone convergence

    419 00:11:55,840 –> 00:11:56,369 theorem.

    420 00:11:56,429 –> 00:11:58,150 And I hope you learned

    421 00:11:58,159 –> 00:11:59,460 how one can prove it.

    422 00:11:59,549 –> 00:12:01,179 And more importantly, you

    423 00:12:01,190 –> 00:12:02,969 now learned when you can

    424 00:12:02,979 –> 00:12:04,700 apply it, you

    425 00:12:04,710 –> 00:12:06,169 need a monotonically

    426 00:12:06,179 –> 00:12:07,559 increasing sequence of

    427 00:12:07,570 –> 00:12:09,109 functions that are non-

    428 00:12:09,119 –> 00:12:09,900 negative.

    429 00:12:10,679 –> 00:12:12,150 Later, we will see how we

    430 00:12:12,159 –> 00:12:13,909 can weaken a little bit this

    431 00:12:13,919 –> 00:12:14,590 assumption.

    432 00:12:15,659 –> 00:12:17,369 However, for the moment this

    433 00:12:17,380 –> 00:12:18,969 version here is powerful

    434 00:12:18,979 –> 00:12:19,409 enough.

    435 00:12:21,299 –> 00:12:22,869 In the following next videos

    436 00:12:22,880 –> 00:12:24,489 we will talk about other

    437 00:12:24,500 –> 00:12:26,059 convergence theorem that are

    438 00:12:26,070 –> 00:12:27,349 also very important.

    439 00:12:28,390 –> 00:12:29,890 I hope I see you there and

    440 00:12:29,900 –> 00:12:31,500 of course, I wish you a nice

    441 00:12:31,510 –> 00:12:31,929 day.

    442 00:12:32,229 –> 00:12:32,849 Bye.

  • Quiz Content

    Q1: Let $(X, \mathcal{A}, \mu)$ be a measure space. The monotone convergence theorem tells us something about a sequence of functions. What is not a premise of this theorem?

    A1: $f_1 \leq f_2 \leq f_3 \leq \cdots$ holds $\mu\text{-a.e}$.

    A2: $f_n: X \rightarrow [0,\infty)$ is measurable.

    A3: $\int_X |f_n| , d\mu < \infty$.

    A4: $\lim_{n \rightarrow \infty} f_n(x) = f(x)$ holds $\mu\text{-a.e}$.

    Q2: Let $(X, \mathcal{A}, \mu)$ be a measure space. Is the following implication correct? $$ f_1 \leq f_2 \leq f_3 \leq \cdots ~ \mu\text{-a.e}$$ $$~\Rightarrow~$$ $$\int_X f_1 \leq \int_X f_2 \leq \int_X f_3 \leq \cdots $$

    A1: Yes

    A2: No

    Q3: Let $(X, \mathcal{A}, \mu)$ be a measure space. If for all simple functions $h$ with $h \leq f$ we have $$ 5 \geq \int_X h , d \mu , ,$$ then we also have $$ 5 \geq \int_X f , d \mu , .$$

    A1: Yes, the statement is correct.

    A2: No, this statement is not correct.

    Q4: Let $(X, \mathcal{A}, \mu)$ be a measure space. If you have positive functions $g_n$ that are measurable. Is the following implication correct? $$ \int_X \sum_{n=1}^\infty g_n , d \mu ~\Rightarrow~ \sum_{n=1}^\infty \int_X g_n , d \mu$$

    A1: Yes, the statement is correct.

    A2: No, this statement is not correct.

  • Back to overview page