• Title: Measurable Maps

  • Series: Measure Theory

  • YouTube-Title: Measure Theory 5 | Measurable Maps

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    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.

  • Quiz Content

    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.

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