• Title: Lebesgue’s Dominated Convergence Theorem

  • Series: Measure Theory

  • YouTube-Title: Measure Theory 10 | Lebesgue’s Dominated Convergence Theorem

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    1 00:00:00,779 –> 00:00:02,599 Hello and welcome back

    2 00:00:02,609 –> 00:00:04,500 to measure theory first.

    3 00:00:04,510 –> 00:00:06,449 As always, let me thank all

    4 00:00:06,460 –> 00:00:08,010 the nice supporters on

    5 00:00:08,020 –> 00:00:08,619 Steady.

    6 00:00:09,560 –> 00:00:11,380 I am very happy to see that

    7 00:00:11,390 –> 00:00:12,899 we have already reached

    8 00:00:12,909 –> 00:00:14,229 part 10.

    9 00:00:14,779 –> 00:00:16,520 And today we will do my

    10 00:00:16,530 –> 00:00:17,780 favorite theory in the

    11 00:00:17,840 –> 00:00:19,379 integration theory.

    12 00:00:20,159 –> 00:00:21,270 And this is Lebesgue’s

    13 00:00:21,629 –> 00:00:23,440 dominated convergence

    14 00:00:23,450 –> 00:00:24,110 theorem.

    15 00:00:24,450 –> 00:00:25,879 Indeed, the name is quite

    16 00:00:25,889 –> 00:00:27,860 fitting because it is as

    17 00:00:27,870 –> 00:00:29,780 before a convergence theorem

    18 00:00:30,530 –> 00:00:31,909 and something has to

    19 00:00:31,920 –> 00:00:33,409 dominate the given

    20 00:00:33,419 –> 00:00:33,990 function.

    21 00:00:34,979 –> 00:00:36,639 Recall a convergence

    22 00:00:36,650 –> 00:00:38,509 theorem tells you when

    23 00:00:38,520 –> 00:00:40,340 you can pull in a

    24 00:00:40,349 –> 00:00:41,680 limit into the

    25 00:00:41,689 –> 00:00:42,360 integral.

    26 00:00:43,099 –> 00:00:44,610 It turns out that this

    27 00:00:44,619 –> 00:00:45,700 convergence theorem that

    28 00:00:45,709 –> 00:00:47,560 is named after Lebesgue is

    29 00:00:47,569 –> 00:00:49,520 very useful and you can apply

    30 00:00:49,529 –> 00:00:50,319 it very often

    31 00:00:51,540 –> 00:00:53,080 before I state the theorem,

    32 00:00:53,090 –> 00:00:54,630 let me first start by

    33 00:00:54,639 –> 00:00:56,520 introducing some notations.

    34 00:00:57,439 –> 00:00:59,240 As always we choose a measure

    35 00:00:59,250 –> 00:00:59,560 space.

    36 00:00:59,569 –> 00:01:01,380 So set X and

    37 00:01:01,389 –> 00:01:03,299 Sigma algebra A and a measure

    38 00:01:03,310 –> 00:01:03,689 mu.

    39 00:01:04,440 –> 00:01:06,360 Now we define a set of Lebesgue-

    40 00:01:07,199 –> 00:01:08,199 integrable functions.

    41 00:01:08,209 –> 00:01:10,019 Therefore, we use this curved

    42 00:01:10,029 –> 00:01:11,680 L most of the

    43 00:01:11,690 –> 00:01:13,540 time the set X and the Sigma

    44 00:01:13,550 –> 00:01:15,230 algebra are so fixed in the

    45 00:01:15,239 –> 00:01:16,730 context that only the

    46 00:01:16,739 –> 00:01:18,209 measure can vary.

    47 00:01:18,639 –> 00:01:20,610 And therefore, I just use

    48 00:01:20,620 –> 00:01:22,089 mu here in a notation.

    49 00:01:23,050 –> 00:01:24,150 In other cases, you would

    50 00:01:24,160 –> 00:01:25,779 write the whole measure space

    51 00:01:25,790 –> 00:01:27,720 here inside. The set

    52 00:01:27,730 –> 00:01:29,669 is now defined as

    53 00:01:29,870 –> 00:01:31,459 the set of all Lebesgue-

    54 00:01:31,830 –> 00:01:32,809 integrable functions.

    55 00:01:33,480 –> 00:01:34,540 Now the functions should

    56 00:01:34,550 –> 00:01:36,500 be defined on

    57 00:01:36,510 –> 00:01:38,419 X and have values in

    58 00:01:38,430 –> 00:01:39,000 R.

    59 00:01:39,449 –> 00:01:40,739 You could also generalize

    60 00:01:40,750 –> 00:01:41,830 that to complex

    61 00:01:41,839 –> 00:01:43,309 values in the end.

    62 00:01:43,319 –> 00:01:45,010 But that’s not so hard indeed.

    63 00:01:45,440 –> 00:01:46,620 The important thing however

    64 00:01:46,629 –> 00:01:47,699 is that they are

    65 00:01:47,709 –> 00:01:48,650 measurable.

    66 00:01:49,730 –> 00:01:51,550 Now, remember we defined the

    67 00:01:51,559 –> 00:01:53,180 integral for

    68 00:01:53,230 –> 00:01:54,800 non-negative functions.

    69 00:01:55,279 –> 00:01:57,019 And for that reason, I look

    70 00:01:57,029 –> 00:01:58,989 at the integral for the absolute

    71 00:01:59,000 –> 00:02:00,930 value of F, this is a

    72 00:02:00,940 –> 00:02:02,150 non-negative function and

    73 00:02:02,160 –> 00:02:03,300 we know it’s still

    74 00:02:03,309 –> 00:02:04,269 measurable.

    75 00:02:04,279 –> 00:02:06,250 So we can look at the integral,

    76 00:02:07,080 –> 00:02:08,979 we know this exists in all

    77 00:02:08,990 –> 00:02:10,899 cases, but in the worst case,

    78 00:02:10,910 –> 00:02:12,139 it could be infinity.

    79 00:02:12,880 –> 00:02:14,419 Hence integrable in this

    80 00:02:14,429 –> 00:02:16,229 sense means it’s not

    81 00:02:16,240 –> 00:02:17,960 infinity, which means

    82 00:02:17,970 –> 00:02:19,779 less than infinity.

    83 00:02:21,139 –> 00:02:21,559 OK.

    84 00:02:21,570 –> 00:02:22,770 So this is the important

    85 00:02:22,779 –> 00:02:24,660 set here, the set of

    86 00:02:24,669 –> 00:02:26,429 Lebesgue-integrable functions

    87 00:02:26,440 –> 00:02:27,440 in this sense,

    88 00:02:28,279 –> 00:02:29,630 maybe I should give you a

    89 00:02:29,639 –> 00:02:31,059 small remark here,

    90 00:02:31,509 –> 00:02:33,259 totally unnecessary in this

    91 00:02:33,270 –> 00:02:33,970 context.

    92 00:02:33,979 –> 00:02:35,080 But it will be important

    93 00:02:35,089 –> 00:02:36,479 later there.

    94 00:02:36,490 –> 00:02:37,839 You will be interested in

    95 00:02:37,850 –> 00:02:39,490 the power for which the

    96 00:02:39,500 –> 00:02:40,350 function is integrable

    97 00:02:41,259 –> 00:02:42,419 this means that you have

    98 00:02:42,429 –> 00:02:44,169 here, the exponent, the power

    99 00:02:44,179 –> 00:02:44,679 one.

    100 00:02:44,800 –> 00:02:46,660 And you also put it on

    101 00:02:46,669 –> 00:02:47,559 the L

    102 00:02:48,589 –> 00:02:49,910 in short, you then call it

    103 00:02:49,919 –> 00:02:51,360 just the L one

    104 00:02:51,369 –> 00:02:53,350 space for such

    105 00:02:53,360 –> 00:02:53,789 functions.

    106 00:02:53,800 –> 00:02:55,699 You can also define the integral

    107 00:02:55,710 –> 00:02:57,279 just by looking at the

    108 00:02:57,289 –> 00:02:58,470 positive and the negative

    109 00:02:58,479 –> 00:03:00,229 part here separately.

    110 00:03:00,710 –> 00:03:02,600 This means that for F in

    111 00:03:02,610 –> 00:03:03,970 our L one

    112 00:03:03,979 –> 00:03:04,740 space,

    113 00:03:06,660 –> 00:03:08,570 we write the function F

    114 00:03:08,580 –> 00:03:10,440 as the combination of two

    115 00:03:10,449 –> 00:03:12,210 non-negative functions and

    116 00:03:12,220 –> 00:03:14,039 I call this first one F

    117 00:03:14,050 –> 00:03:15,869 plus and the second one

    118 00:03:15,899 –> 00:03:16,910 F minus

    119 00:03:17,759 –> 00:03:19,179 and the combination is given

    120 00:03:19,190 –> 00:03:20,570 by a minus sign.

    121 00:03:20,820 –> 00:03:22,539 And, and the idea is F

    122 00:03:22,550 –> 00:03:24,199 plus F minus

    123 00:03:24,380 –> 00:03:25,889 are non-negative.

    124 00:03:27,039 –> 00:03:27,339 OK.

    125 00:03:27,350 –> 00:03:29,199 Maybe a short picture for

    126 00:03:29,210 –> 00:03:29,690 this.

    127 00:03:31,039 –> 00:03:32,699 If this is the

    128 00:03:32,710 –> 00:03:34,500 graph of the function F,

    129 00:03:36,339 –> 00:03:38,320 then this part here

    130 00:03:38,600 –> 00:03:40,440 is the graph of our function

    131 00:03:40,449 –> 00:03:41,669 F plus.

    132 00:03:45,339 –> 00:03:47,080 And of course, this part

    133 00:03:47,089 –> 00:03:48,889 here then is

    134 00:03:48,899 –> 00:03:50,619 the graph corresponding

    135 00:03:50,630 –> 00:03:51,869 to F minus.

    136 00:03:51,880 –> 00:03:53,500 It’s not exactly F minus,

    137 00:03:53,509 –> 00:03:54,550 this would be

    138 00:03:54,910 –> 00:03:56,460 minus F

    139 00:03:56,470 –> 00:03:57,080 minus.

    140 00:03:58,080 –> 00:04:00,029 However, obviously adding

    141 00:04:00,039 –> 00:04:01,669 both these functions gives

    142 00:04:01,679 –> 00:04:03,429 you back our original F.

    143 00:04:04,699 –> 00:04:06,570 Now you may immediately believe

    144 00:04:06,580 –> 00:04:08,279 me that for all functions

    145 00:04:08,289 –> 00:04:09,970 F, you can split them up

    146 00:04:09,979 –> 00:04:11,399 into these two parts

    147 00:04:12,089 –> 00:04:13,740 in the positive part above

    148 00:04:13,750 –> 00:04:14,690 the X axis.

    149 00:04:14,699 –> 00:04:16,428 And the negative part below

    150 00:04:16,440 –> 00:04:17,329 the X axis,

    151 00:04:18,558 –> 00:04:20,269 we have to do this because

    152 00:04:20,278 –> 00:04:22,058 we only define the integral

    153 00:04:22,069 –> 00:04:23,898 for non-negative functions.

    154 00:04:23,908 –> 00:04:25,899 As you remember, the

    155 00:04:25,910 –> 00:04:27,299 idea is then I have an

    156 00:04:27,309 –> 00:04:29,140 integral for F plus

    157 00:04:29,269 –> 00:04:31,149 and also an integral for

    158 00:04:31,160 –> 00:04:32,209 F minus.

    159 00:04:32,450 –> 00:04:34,279 And then I will subtract

    160 00:04:34,290 –> 00:04:36,149 the areas then I

    161 00:04:36,160 –> 00:04:37,809 don’t get out the area

    162 00:04:37,980 –> 00:04:39,940 but I get out an orientated

    163 00:04:39,950 –> 00:04:41,809 area where I

    164 00:04:41,820 –> 00:04:43,450 subtract the parts that

    165 00:04:43,459 –> 00:04:45,269 lie below the X axis.

    166 00:04:46,160 –> 00:04:47,359 This is the integral notion

    167 00:04:47,369 –> 00:04:48,100 we want.

    168 00:04:48,109 –> 00:04:49,660 And we also have this for

    169 00:04:49,670 –> 00:04:50,809 the Riemann integral, as

    170 00:04:50,820 –> 00:04:51,579 you remember.

    171 00:04:52,440 –> 00:04:54,420 Hence, therefore, the definition

    172 00:04:54,429 –> 00:04:56,209 is given as the

    173 00:04:56,220 –> 00:04:58,140 following symbol, the integral

    174 00:04:58,149 –> 00:04:59,940 over X for the

    175 00:04:59,950 –> 00:05:01,739 function F dmu

    176 00:05:02,070 –> 00:05:03,630 defined as

    177 00:05:04,320 –> 00:05:05,600 the well-defined integral

    178 00:05:05,609 –> 00:05:06,700 of F plus

    179 00:05:07,369 –> 00:05:09,339 has a non-negative function minus

    180 00:05:10,320 –> 00:05:11,700 The well-defined value

    181 00:05:12,200 –> 00:05:13,559 for the integral F minus,

    182 00:05:15,089 –> 00:05:16,970 both parts have a

    183 00:05:16,980 –> 00:05:18,190 positive value.

    184 00:05:18,329 –> 00:05:19,839 And we also know it’s

    185 00:05:19,850 –> 00:05:21,709 finite by this

    186 00:05:21,720 –> 00:05:22,589 assumption here.

    187 00:05:23,630 –> 00:05:24,709 Therefore, the subtraction

    188 00:05:24,720 –> 00:05:25,890 is no problem at all.

    189 00:05:25,899 –> 00:05:27,450 We get out a real number

    190 00:05:27,459 –> 00:05:27,970 in the end.

    191 00:05:28,640 –> 00:05:29,260 OK.

    192 00:05:29,269 –> 00:05:30,839 Now you know what the integral

    193 00:05:30,850 –> 00:05:32,359 symbol is for measurable

    194 00:05:32,369 –> 00:05:34,250 functions with real values.

    195 00:05:35,390 –> 00:05:37,029 I skipped some details for

    196 00:05:37,040 –> 00:05:38,179 the definition of F plus

    197 00:05:38,190 –> 00:05:39,970 and F minus because I think

    198 00:05:39,980 –> 00:05:41,160 the picture is sufficient

    199 00:05:41,170 –> 00:05:41,410 here.

    200 00:05:42,440 –> 00:05:42,829 OK.

    201 00:05:42,839 –> 00:05:44,200 Now I can finally

    202 00:05:44,209 –> 00:05:45,619 state Lebesgue’s

    203 00:05:45,630 –> 00:05:47,619 dominated convergence theorem.

    204 00:05:48,619 –> 00:05:50,170 What we need here is a

    205 00:05:50,179 –> 00:05:52,079 sequence of functions and

    206 00:05:52,089 –> 00:05:54,059 I call it FN defined on

    207 00:05:54,070 –> 00:05:55,670 X and they can

    208 00:05:55,679 –> 00:05:57,559 have real values now

    209 00:05:58,070 –> 00:05:59,079 and of course, they should

    210 00:05:59,089 –> 00:06:00,390 be measurable.

    211 00:06:01,309 –> 00:06:03,230 Of course, you can visualize

    212 00:06:03,239 –> 00:06:05,070 this always as

    213 00:06:05,079 –> 00:06:06,769 sequence of graphs.

    214 00:06:06,839 –> 00:06:08,799 So this would be F one.

    215 00:06:09,250 –> 00:06:11,119 And then the next thing would

    216 00:06:11,130 –> 00:06:12,910 be here as F two

    217 00:06:15,420 –> 00:06:17,019 and then here at F three

    218 00:06:17,029 –> 00:06:18,279 and so on.

    219 00:06:19,000 –> 00:06:20,899 For such a sequence of functions

    220 00:06:20,910 –> 00:06:22,500 you can always ask

    221 00:06:22,510 –> 00:06:24,260 about the pointwise limit.

    222 00:06:25,130 –> 00:06:26,959 This means that you fix a

    223 00:06:26,970 –> 00:06:28,649 point X on the X

    224 00:06:28,660 –> 00:06:30,390 axis and

    225 00:06:30,399 –> 00:06:32,100 look at the values for the

    226 00:06:32,109 –> 00:06:32,589 functions.

    227 00:06:32,600 –> 00:06:34,350 So you have one value here,

    228 00:06:34,609 –> 00:06:35,910 the next one here.

    229 00:06:36,109 –> 00:06:37,339 So you get out a

    230 00:06:37,350 –> 00:06:39,230 normal sequence of

    231 00:06:39,239 –> 00:06:40,170 real numbers.

    232 00:06:40,859 –> 00:06:42,559 Therefore, you can ask what

    233 00:06:42,570 –> 00:06:44,079 happens with this

    234 00:06:44,089 –> 00:06:45,519 normal sequence of

    235 00:06:45,529 –> 00:06:46,160 numbers.

    236 00:06:47,519 –> 00:06:49,209 If it is convergent, you

    237 00:06:49,220 –> 00:06:51,140 get out a limit which is

    238 00:06:51,149 –> 00:06:52,859 then a point here.

    239 00:06:54,149 –> 00:06:55,579 In the case, you can do this

    240 00:06:55,589 –> 00:06:57,390 for all X here, you get

    241 00:06:57,399 –> 00:06:58,869 out a lot of points here.

    242 00:06:58,880 –> 00:07:00,019 And indeed, you get out a

    243 00:07:00,029 –> 00:07:01,720 new graph and therefore a

    244 00:07:01,730 –> 00:07:02,720 new function.

    245 00:07:03,890 –> 00:07:05,200 And this is the pointwise

    246 00:07:05,209 –> 00:07:06,309 limit function.

    247 00:07:06,320 –> 00:07:08,070 And we will call it just

    248 00:07:08,079 –> 00:07:08,750 F here.

    249 00:07:09,470 –> 00:07:11,070 And this is what I also want

    250 00:07:11,079 –> 00:07:12,890 to put into the assumptions

    251 00:07:12,899 –> 00:07:14,170 of our theorem here.

    252 00:07:15,010 –> 00:07:16,500 This means that we also have

    253 00:07:16,510 –> 00:07:17,750 our function F here

    254 00:07:17,809 –> 00:07:19,000 also with the

    255 00:07:19,010 –> 00:07:19,890 values.

    256 00:07:19,989 –> 00:07:21,950 And the following property,

    257 00:07:23,390 –> 00:07:25,339 if we fix a point X

    258 00:07:25,480 –> 00:07:26,859 for all our functions

    259 00:07:26,869 –> 00:07:28,709 FN, then this should

    260 00:07:28,720 –> 00:07:30,579 be convergent to

    261 00:07:30,589 –> 00:07:31,809 F of X.

    262 00:07:33,549 –> 00:07:34,890 And I told you we want this

    263 00:07:34,899 –> 00:07:36,850 property for all

    264 00:07:36,859 –> 00:07:38,570 X on the X axis.

    265 00:07:38,589 –> 00:07:40,209 So for all lower

    266 00:07:40,220 –> 00:07:42,200 case X in our capital

    267 00:07:42,209 –> 00:07:44,079 X, however,

    268 00:07:44,089 –> 00:07:45,679 you know, we are in the realm

    269 00:07:45,690 –> 00:07:46,989 of measure theory,

    270 00:07:47,660 –> 00:07:49,250 this means we don’t need

    271 00:07:49,260 –> 00:07:50,959 this property exactly

    272 00:07:50,970 –> 00:07:51,700 everywhere.

    273 00:07:52,220 –> 00:07:53,690 It’s sufficient that we have

    274 00:07:53,700 –> 00:07:55,440 it almost everywhere,

    275 00:07:56,429 –> 00:07:58,399 almost always means

    276 00:07:58,410 –> 00:08:00,019 with respect to our measure

    277 00:08:00,029 –> 00:08:00,709 mu here.

    278 00:08:01,549 –> 00:08:03,160 And in short, we write this

    279 00:08:03,170 –> 00:08:04,380 as mu

    280 00:08:04,730 –> 00:08:06,440 almost everywhere.

    281 00:08:07,829 –> 00:08:09,589 Please recall that this

    282 00:08:09,600 –> 00:08:11,299 means exactly that the

    283 00:08:11,309 –> 00:08:13,170 set where this does

    284 00:08:13,179 –> 00:08:14,709 not hold is a

    285 00:08:14,720 –> 00:08:16,040 set of measure

    286 00:08:16,049 –> 00:08:16,910 zero.

    287 00:08:17,390 –> 00:08:19,369 So mu of the set

    288 00:08:19,380 –> 00:08:20,809 is equal to zero.

    289 00:08:21,839 –> 00:08:22,420 OK.

    290 00:08:22,529 –> 00:08:24,250 So until now the assumptions

    291 00:08:24,260 –> 00:08:25,420 are not so strange,

    292 00:08:26,440 –> 00:08:28,299 you have a sequence of measurable

    293 00:08:28,309 –> 00:08:30,100 functions and also

    294 00:08:30,149 –> 00:08:32,000 the pointwise limit of

    295 00:08:32,010 –> 00:08:33,010 this sequence.

    296 00:08:33,869 –> 00:08:35,169 And now you could ask a lot

    297 00:08:35,179 –> 00:08:36,669 of questions are the

    298 00:08:36,679 –> 00:08:38,390 functions integrable?

    299 00:08:38,400 –> 00:08:40,169 So do they lie in our

    300 00:08:40,179 –> 00:08:41,289 L one space.

    301 00:08:41,880 –> 00:08:43,419 And if they do, what about

    302 00:08:43,429 –> 00:08:44,859 our point was estimate function

    303 00:08:44,869 –> 00:08:46,770 F then and more

    304 00:08:46,780 –> 00:08:48,179 importantly, can I

    305 00:08:48,190 –> 00:08:49,609 swap the limit

    306 00:08:49,619 –> 00:08:51,549 And the integral? Or in

    307 00:08:51,559 –> 00:08:53,349 other words, can I pull

    308 00:08:53,359 –> 00:08:54,909 in the limit into the

    309 00:08:54,919 –> 00:08:55,549 integral?

    310 00:08:56,450 –> 00:08:57,849 Now Lebesgue’s dominated

    311 00:08:57,859 –> 00:08:59,669 convergence theorem says

    312 00:08:59,679 –> 00:09:01,469 yes to all of these

    313 00:09:01,479 –> 00:09:02,309 questions,

    314 00:09:02,780 –> 00:09:04,619 if we add one

    315 00:09:04,630 –> 00:09:05,750 more assumption

    316 00:09:06,539 –> 00:09:08,479 and this is where the dominated

    317 00:09:08,489 –> 00:09:09,599 comes into the play,

    318 00:09:10,669 –> 00:09:11,869 we have that the absolute

    319 00:09:11,880 –> 00:09:13,650 value of all these

    320 00:09:13,659 –> 00:09:15,039 functions is

    321 00:09:15,049 –> 00:09:16,900 bounded by a function

    322 00:09:16,909 –> 00:09:17,369 G.

    323 00:09:18,080 –> 00:09:19,030 Of course, you should read

    324 00:09:19,039 –> 00:09:20,320 this point wisely.

    325 00:09:20,330 –> 00:09:22,190 So if you put in an X

    326 00:09:22,200 –> 00:09:24,049 here, this inequality

    327 00:09:24,059 –> 00:09:25,679 holds the

    328 00:09:25,690 –> 00:09:27,130 function G now has the

    329 00:09:27,140 –> 00:09:28,520 property that it is

    330 00:09:28,530 –> 00:09:30,260 integrable, which means it

    331 00:09:30,270 –> 00:09:31,849 comes from our L one

    332 00:09:31,859 –> 00:09:32,479 space

    333 00:09:34,059 –> 00:09:35,679 and obviously it should be

    334 00:09:35,690 –> 00:09:37,419 the same G for

    335 00:09:37,429 –> 00:09:38,159 all N.

    336 00:09:41,450 –> 00:09:43,440 Now this G is what one

    337 00:09:43,450 –> 00:09:45,419 usually calls an integrable

    338 00:09:45,859 –> 00:09:46,909 majorant.

    339 00:09:47,380 –> 00:09:48,520 So this is what Lebesgue’s

    340 00:09:48,770 –> 00:09:50,619 dominated convergence theorem

    341 00:09:50,630 –> 00:09:52,570 indeed needs it needs an

    342 00:09:52,580 –> 00:09:54,250 integrable majorant.

    343 00:09:54,679 –> 00:09:56,280 So a function that

    344 00:09:56,289 –> 00:09:58,229 lies above all

    345 00:09:58,239 –> 00:09:59,469 the other functions here.

    346 00:10:00,590 –> 00:10:01,929 This means it’s not important

    347 00:10:01,940 –> 00:10:03,349 what exactly the function

    348 00:10:03,359 –> 00:10:04,130 G is.

    349 00:10:04,140 –> 00:10:05,479 You only need this

    350 00:10:05,489 –> 00:10:07,250 inequality and you

    351 00:10:07,260 –> 00:10:09,049 need that it’s integrable

    352 00:10:09,059 –> 00:10:10,969 from these two properties.

    353 00:10:10,979 –> 00:10:12,739 Now, all the other things

    354 00:10:12,750 –> 00:10:14,500 follow, for

    355 00:10:14,510 –> 00:10:16,179 example, if you know

    356 00:10:16,210 –> 00:10:17,719 that G has a finite

    357 00:10:17,729 –> 00:10:19,630 integral, then all the

    358 00:10:19,640 –> 00:10:21,020 other functions should be

    359 00:10:21,030 –> 00:10:21,960 also integrable.

    360 00:10:23,489 –> 00:10:23,950 OK.

    361 00:10:23,960 –> 00:10:25,710 So let’s write that down

    362 00:10:25,719 –> 00:10:27,630 as the implication of the

    363 00:10:27,640 –> 00:10:28,190 theorem,

    364 00:10:30,960 –> 00:10:32,539 all the functions F

    365 00:10:32,559 –> 00:10:34,520 one F two

    366 00:10:34,549 –> 00:10:36,500 F three and so on

    367 00:10:36,669 –> 00:10:38,299 lie in our

    368 00:10:38,380 –> 00:10:39,979 L one space.

    369 00:10:41,989 –> 00:10:43,729 And we can say the same

    370 00:10:43,890 –> 00:10:45,380 for our point west limit

    371 00:10:45,390 –> 00:10:46,239 function F.

    372 00:10:47,159 –> 00:10:48,809 Of course, we know it should

    373 00:10:48,820 –> 00:10:50,580 be measurable as a limit,

    374 00:10:50,809 –> 00:10:52,469 but it’s also

    375 00:10:52,760 –> 00:10:53,099 integrable.

    376 00:10:54,099 –> 00:10:55,369 Now let’s look at the integrals

    377 00:10:55,469 –> 00:10:56,820 we know

    378 00:10:57,229 –> 00:10:59,179 for all FN, the

    379 00:10:59,190 –> 00:11:00,760 integral will exist

    380 00:11:01,179 –> 00:11:02,710 and defines a real

    381 00:11:02,719 –> 00:11:03,239 number.

    382 00:11:04,119 –> 00:11:05,390 Therefore, we have here a

    383 00:11:05,400 –> 00:11:07,210 new sequence of real numbers

    384 00:11:07,219 –> 00:11:09,140 and we can ask what is the

    385 00:11:09,150 –> 00:11:09,700 limit?

    386 00:11:11,580 –> 00:11:13,039 And the answer is you can

    387 00:11:13,049 –> 00:11:14,200 pull the limit

    388 00:11:14,210 –> 00:11:16,000 inside and there we

    389 00:11:16,010 –> 00:11:17,440 have our pointwise limit

    390 00:11:17,450 –> 00:11:18,219 of f_n.

    391 00:11:18,229 –> 00:11:19,799 So this is what we called

    392 00:11:19,809 –> 00:11:20,359 F

    393 00:11:22,010 –> 00:11:23,469 and that’s now Lebesgue’s

    394 00:11:23,479 –> 00:11:24,729 dominated convergence theorem

    395 00:11:26,070 –> 00:11:27,700 as a convergence theorem,

    396 00:11:27,710 –> 00:11:29,669 it tells you when you are

    397 00:11:29,679 –> 00:11:31,549 allowed to push the limit

    398 00:11:31,719 –> 00:11:33,130 inside the integral.

    399 00:11:34,159 –> 00:11:36,099 And here you see, you just

    400 00:11:36,109 –> 00:11:37,690 have to find a suitable

    401 00:11:37,700 –> 00:11:39,559 function G nothing more.

    402 00:11:40,440 –> 00:11:41,919 And that is the reason why

    403 00:11:41,929 –> 00:11:43,250 this theorem is so

    404 00:11:43,260 –> 00:11:45,099 useful and you can apply

    405 00:11:45,109 –> 00:11:45,979 it very often.

    406 00:11:47,210 –> 00:11:47,590 OK?

    407 00:11:47,599 –> 00:11:49,210 I hope you’re also intrigued

    408 00:11:49,219 –> 00:11:51,119 by the theorem as much as I

    409 00:11:51,130 –> 00:11:52,830 am and that you will

    410 00:11:52,840 –> 00:11:54,169 watch the next video in the

    411 00:11:54,179 –> 00:11:54,830 series

    412 00:11:55,700 –> 00:11:57,340 where I of course show

    413 00:11:57,349 –> 00:11:59,099 you the whole proof of

    414 00:11:59,109 –> 00:12:00,820 this dominated convergence

    415 00:12:00,830 –> 00:12:01,309 theorem.

    416 00:12:02,239 –> 00:12:04,200 And after this, I can show

    417 00:12:04,210 –> 00:12:06,020 you a lot of examples

    418 00:12:06,030 –> 00:12:07,950 where Lebesgue’s dominated convergence

    419 00:12:07,960 –> 00:12:09,780 theorem is very important.

    420 00:12:10,520 –> 00:12:10,869 OK.

    421 00:12:10,880 –> 00:12:12,549 Thank you very much for listening

    422 00:12:12,559 –> 00:12:13,789 and see you next time.

    423 00:12:13,960 –> 00:12:14,590 Bye.

  • Quiz Content

    Q1: Let $X$ be a set. What does $f^+: X \rightarrow \mathbb{R} $ for a function $f: X \rightarrow \mathbb{R}$ mean?

    A1: $f^+(x) := \begin{cases} f(x) , &, ~ \mbox{if } f(x) > 0 \ 0 , &, ~ \mbox{else } \end{cases}$

    A2: $f^+(x) := \begin{cases} f(x) , &, ~ \mbox{if } f(x) < 0 \ 0 , &, ~ \mbox{else } \end{cases}$

    A3: $f^+(x) := \begin{cases} f(x) , &, ~ \mbox{if } f(x) = 0 \ 0 , &, ~ \mbox{else } \end{cases}$

    A4: $f^+(x) := 2 \cdot f(x)$

    Q2: Let $X$ be a set. What does $f^-: X \rightarrow \mathbb{R} $ for a function $f: X \rightarrow \mathbb{R}$ mean?

    A1: $f^-(x) := \begin{cases}- f(x) , &, ~ \mbox{if } f(x) < 0 \ 0 , &, ~ \mbox{else } \end{cases}$

    A2: $f^-(x) := \begin{cases} f(x) , &, ~ \mbox{if } f(x) < 0 \ 0 , &, ~ \mbox{else } \end{cases}$

    A3: $f^-(x) := \begin{cases} - f(x) , &, ~ \mbox{if } f(x) > 0 \ 0 , &, ~ \mbox{else } \end{cases}$

    A4: $f^-(x) := \begin{cases} f(x) , &, ~ \mbox{if } f(x) > 0 \ -f(x) , &, ~ \mbox{else } \end{cases}$

    Q3: Let $(X, \mathcal{A}, \mu)$ be a measure space and $f_n: X \rightarrow \mathbb{R}$ be measurable for all $n \in \mathbb{N}$ and $f: X \rightarrow \mathbb{R}$. What is not an assumption of Lebesgue’s dominated convergence theorem?

    A1: $(f_n)$ is uniformly convergent to $f$.

    A2: $f_n(x) \xrightarrow{n \rightarrow \infty} f(x)$ for $x \in X$ $\mu$-a.e.

    A3: There is a $g \in \mathcal{L}(\mu)$ such that $|f_n| \leq g$ for all $n \in \mathbb{N}$.

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