1 00:00:00,579 –> 00:00:02,170 Hello and welcome

2 00:00:02,180 –> 00:00:02,720 back.

3 00:00:02,730 –> 00:00:04,519 First, let me thank all the

4 00:00:04,530 –> 00:00:06,219 nice people that support

5 00:00:06,230 –> 00:00:07,760 this channel on Steady.

6 00:00:08,850 –> 00:00:10,539 And now let’s continue with

7 00:00:10,550 –> 00:00:12,439 measure theory namely

8 00:00:12,449 –> 00:00:14,069 with part six.

9 00:00:15,989 –> 00:00:17,420 And we will finally talk

10 00:00:17,430 –> 00:00:19,190 about the Lebesgue integral but

11 00:00:19,200 –> 00:00:20,639 first about the Lebesgue

12 00:00:20,649 –> 00:00:22,399 integral of so

13 00:00:22,409 –> 00:00:24,280 called-step functions.

14 00:00:25,090 –> 00:00:26,920 So we will learn how to

15 00:00:26,930 –> 00:00:28,799 integrate functions that

16 00:00:28,809 –> 00:00:30,120 are defined on an

17 00:00:30,129 –> 00:00:31,840 abstract measure space.

18 00:00:33,020 –> 00:00:34,619 As a short recap, a

19 00:00:34,630 –> 00:00:36,240 measure space is

20 00:00:36,250 –> 00:00:37,709 nothing more than a

21 00:00:37,720 –> 00:00:38,400 triple.

22 00:00:39,110 –> 00:00:39,330 There.

23 00:00:39,340 –> 00:00:40,779 We have a set X,

24 00:00:41,209 –> 00:00:42,770 a sigma algebra A

25 00:00:42,930 –> 00:00:44,470 and also a measure

26 00:00:44,479 –> 00:00:45,009 mu.

27 00:00:45,830 –> 00:00:47,779 This means that X could

28 00:00:47,790 –> 00:00:49,080 be any set

29 00:00:50,049 –> 00:00:51,979 but A is a special

30 00:00:51,990 –> 00:00:53,580 collection of subsets of

31 00:00:53,590 –> 00:00:54,049 X.

32 00:00:54,529 –> 00:00:56,189 And the third ingredient,

33 00:00:56,360 –> 00:00:58,040 the measure mu itself is

34 00:00:58,049 –> 00:00:59,750 indeed a map

35 00:01:00,259 –> 00:01:02,139 where the domain is the sigma

36 00:01:02,150 –> 00:01:03,880 algebra A and the co-

37 00:01:03,889 –> 00:01:05,290 domain is the

38 00:01:05,300 –> 00:01:06,980 interval included

39 00:01:06,989 –> 00:01:08,580 zero and also

40 00:01:08,589 –> 00:01:10,220 included the symbol

41 00:01:10,230 –> 00:01:10,910 infinity.

42 00:01:11,860 –> 00:01:13,580 Now, with respect to

43 00:01:13,589 –> 00:01:15,519 this abstract measure

44 00:01:15,529 –> 00:01:16,870 space, we want to

45 00:01:16,879 –> 00:01:18,720 integrate some special

46 00:01:18,730 –> 00:01:19,510 functions.

47 00:01:20,470 –> 00:01:22,139 Indeed what we need are

48 00:01:22,150 –> 00:01:24,089 measurable maps defined in

49 00:01:24,099 –> 00:01:25,970 the last video I

50 00:01:25,980 –> 00:01:27,559 will use the letter F for

51 00:01:27,569 –> 00:01:29,529 such maps that start

52 00:01:29,540 –> 00:01:31,529 with X and go into

53 00:01:31,540 –> 00:01:32,730 the real number line.

54 00:01:33,709 –> 00:01:35,610 Now you should not forget

55 00:01:35,620 –> 00:01:37,519 that we have a Sigma algebra

56 00:01:37,529 –> 00:01:39,089 on the left namely

57 00:01:39,099 –> 00:01:40,970 A and also a Sigma

58 00:01:40,980 –> 00:01:42,150 algebra on the right.

59 00:01:42,580 –> 00:01:44,040 And there we have the Borel

60 00:01:44,309 –> 00:01:45,370 Sigma algebra.

61 00:01:46,650 –> 00:01:48,309 Also recall

62 00:01:48,319 –> 00:01:50,309 that measurable means

63 00:01:50,519 –> 00:01:52,209 that all the preimages

64 00:01:52,220 –> 00:01:53,959 from elements from the sigma

65 00:01:53,970 –> 00:01:55,910 algebra here lie

66 00:01:55,919 –> 00:01:57,230 in the sigma algebra A.

67 00:01:57,239 –> 00:01:59,110 In other

68 00:01:59,120 –> 00:02:00,860 words, preimage

69 00:02:00,870 –> 00:02:02,720 of a set

70 00:02:02,730 –> 00:02:04,599 E is in

71 00:02:04,610 –> 00:02:06,519 A for all

72 00:02:06,529 –> 00:02:08,100 Borel sets E

73 00:02:08,630 –> 00:02:09,139 OK.

74 00:02:09,149 –> 00:02:10,839 So these are the functions

75 00:02:10,850 –> 00:02:12,160 where we want to be able

76 00:02:12,169 –> 00:02:13,940 to integrate them in the

77 00:02:13,949 –> 00:02:14,320 end.

78 00:02:15,160 –> 00:02:16,910 However, at the moment,

79 00:02:17,169 –> 00:02:18,910 they might be too complicated.

80 00:02:18,919 –> 00:02:20,470 So we start with functions

81 00:02:20,479 –> 00:02:21,589 that we already know.

82 00:02:22,339 –> 00:02:24,070 For example, we already

83 00:02:24,080 –> 00:02:25,710 know that the characteristic

84 00:02:25,720 –> 00:02:27,669 function is a measurable

85 00:02:27,679 –> 00:02:28,309 map.

86 00:02:28,320 –> 00:02:30,229 If we choose a measurable

87 00:02:30,240 –> 00:02:30,669 set,

88 00:02:32,050 –> 00:02:33,240 this means A

89 00:02:33,529 –> 00:02:35,350 lies in the Sigma algebra.

90 00:02:36,419 –> 00:02:38,020 We already know how to sketch

91 00:02:38,029 –> 00:02:39,080 this function.

92 00:02:39,880 –> 00:02:41,619 If we have our abstract X

93 00:02:41,630 –> 00:02:43,179 here on the line

94 00:02:43,330 –> 00:02:45,059 and maybe this

95 00:02:45,070 –> 00:02:46,539 is the set A.

96 00:02:46,809 –> 00:02:48,550 So these two things together

97 00:02:48,559 –> 00:02:50,529 are A, then we

98 00:02:50,539 –> 00:02:52,479 can sketch the graph of

99 00:02:52,490 –> 00:02:54,029 our characteristic function.

100 00:02:56,149 –> 00:02:57,820 It will be zero here

101 00:02:58,119 –> 00:02:59,789 and has to value you

102 00:02:59,800 –> 00:03:01,759 one where the set A

103 00:03:01,770 –> 00:03:02,279 lies.

104 00:03:02,289 –> 00:03:04,179 So here it will be one and

105 00:03:04,190 –> 00:03:05,490 here also zero

106 00:03:05,789 –> 00:03:07,720 and there also zero

107 00:03:08,589 –> 00:03:10,160 for visualization of the

108 00:03:10,169 –> 00:03:12,059 integral, it’s always

109 00:03:12,070 –> 00:03:14,000 good to see the integral

110 00:03:14,009 –> 00:03:15,970 as the area below

111 00:03:15,979 –> 00:03:16,779 the graph

112 00:03:16,789 –> 00:03:17,960 And the x axis

113 00:03:18,820 –> 00:03:19,809 here, this would mean we

114 00:03:19,820 –> 00:03:21,550 look at this area

115 00:03:21,559 –> 00:03:22,190 here

116 00:03:23,690 –> 00:03:25,190 and the area here.

117 00:03:29,149 –> 00:03:30,639 And because the value of

118 00:03:30,649 –> 00:03:32,289 the function is just one,

119 00:03:32,300 –> 00:03:34,250 it does not matter how abstract

120 00:03:34,259 –> 00:03:35,330 this whole measure space

121 00:03:35,339 –> 00:03:37,130 is, this area should

122 00:03:37,139 –> 00:03:39,130 be exactly the same as

123 00:03:39,139 –> 00:03:41,009 the volume, the measure

124 00:03:41,020 –> 00:03:42,279 of the set A

125 00:03:43,160 –> 00:03:44,380 or in other words, a

126 00:03:44,389 –> 00:03:46,059 meaningful integral

127 00:03:46,070 –> 00:03:46,740 notion.

128 00:03:46,750 –> 00:03:48,179 And maybe let’s call it just

129 00:03:48,190 –> 00:03:49,330 by capital I

130 00:03:49,889 –> 00:03:51,369 of this characteristic

131 00:03:51,380 –> 00:03:52,899 function should always

132 00:03:52,910 –> 00:03:54,460 fulfill that.

133 00:03:54,470 –> 00:03:56,039 This integral

134 00:03:56,169 –> 00:03:58,029 is the measure of the

135 00:03:58,039 –> 00:03:58,949 set A.

136 00:04:00,089 –> 00:04:01,470 Now we have a new symbol

137 00:04:01,479 –> 00:04:03,020 on the left I of a

138 00:04:03,029 –> 00:04:03,800 function.

139 00:04:03,809 –> 00:04:05,750 So you should read this indeed,

140 00:04:05,770 –> 00:04:07,149 as a definition,

141 00:04:08,520 –> 00:04:10,270 this means now you know

142 00:04:10,279 –> 00:04:12,210 how to integrate characteristic

143 00:04:12,220 –> 00:04:12,740 functions.

144 00:04:13,949 –> 00:04:15,050 In fact, there are other

145 00:04:15,059 –> 00:04:16,329 functions where we can

146 00:04:16,339 –> 00:04:18,079 define the integral

147 00:04:18,089 –> 00:04:19,798 also in such a simple

148 00:04:19,809 –> 00:04:20,190 way.

149 00:04:20,380 –> 00:04:22,269 And these are called simple

150 00:04:22,279 –> 00:04:23,089 functions.

151 00:04:23,950 –> 00:04:25,649 There are also a lot of other

152 00:04:25,660 –> 00:04:27,570 names that one uses for such

153 00:04:27,579 –> 00:04:28,649 simple functions.

154 00:04:29,559 –> 00:04:30,940 For example, step

155 00:04:30,950 –> 00:04:32,470 functions as I

156 00:04:32,480 –> 00:04:34,029 did in the

157 00:04:34,040 –> 00:04:35,640 title of this video

158 00:04:36,059 –> 00:04:36,940 and also

159 00:04:37,000 –> 00:04:38,859 staircase functions

160 00:04:39,649 –> 00:04:41,100 and also other

161 00:04:41,109 –> 00:04:41,839 names.

162 00:04:42,779 –> 00:04:44,489 And you see the name

163 00:04:44,500 –> 00:04:46,359 might not be important.

164 00:04:46,380 –> 00:04:47,809 You just have to know what

165 00:04:47,820 –> 00:04:49,730 such simple functions are.

166 00:04:50,799 –> 00:04:51,429 In short.

167 00:04:51,440 –> 00:04:53,220 A simple function F

168 00:04:53,230 –> 00:04:54,730 is just a linear

169 00:04:54,739 –> 00:04:56,660 combination of such

170 00:04:56,670 –> 00:04:58,140 characteristic functions.

171 00:04:58,660 –> 00:05:00,130 This means you can write

172 00:05:00,140 –> 00:05:01,890 F of X as a

173 00:05:01,899 –> 00:05:02,739 sum.

174 00:05:02,869 –> 00:05:04,170 Where we will start with

175 00:05:04,200 –> 00:05:06,070 one and go to a fixed

176 00:05:06,079 –> 00:05:07,820 N and where we

177 00:05:07,829 –> 00:05:09,679 have numbers C

178 00:05:09,690 –> 00:05:11,149 I and

179 00:05:11,160 –> 00:05:12,649 characteristic functions

180 00:05:12,660 –> 00:05:14,250 corresponding to some

181 00:05:14,260 –> 00:05:16,209 measurable set A

182 00:05:16,220 –> 00:05:16,850 I.

183 00:05:18,619 –> 00:05:19,809 In other words, a

184 00:05:19,829 –> 00:05:21,779 function F is called

185 00:05:21,790 –> 00:05:22,559 simple.

186 00:05:22,570 –> 00:05:23,910 If you can find

187 00:05:23,920 –> 00:05:25,809 measurable sets A_1

188 00:05:25,820 –> 00:05:26,869 till A_n

189 00:05:27,750 –> 00:05:29,390 and real numbers c_1

190 00:05:29,399 –> 00:05:31,200 to c_n

191 00:05:31,380 –> 00:05:32,839 such that you can write the

192 00:05:32,850 –> 00:05:34,600 function in this form

193 00:05:35,500 –> 00:05:36,899 because we know that the

194 00:05:36,910 –> 00:05:38,260 characteristic functions

195 00:05:38,269 –> 00:05:40,100 are measurable and also

196 00:05:40,109 –> 00:05:42,000 sums of measurable

197 00:05:42,010 –> 00:05:43,959 maps are also measurable.

198 00:05:44,059 –> 00:05:45,910 We also know that this

199 00:05:45,920 –> 00:05:47,160 simple function is

200 00:05:47,170 –> 00:05:48,000 therefore

201 00:05:48,019 –> 00:05:49,239 measurable.

202 00:05:50,519 –> 00:05:52,040 Also here, I would

203 00:05:52,049 –> 00:05:53,700 say to get some

204 00:05:53,709 –> 00:05:55,500 visualization, let’s

205 00:05:55,510 –> 00:05:57,049 sketch the graph of such

206 00:05:57,059 –> 00:05:57,899 a simple function.

207 00:05:58,980 –> 00:06:00,709 This is the same as before.

208 00:06:00,720 –> 00:06:02,619 So we have X here and

209 00:06:02,630 –> 00:06:04,540 R here and

210 00:06:04,549 –> 00:06:06,540 maybe here now we have set

211 00:06:06,549 –> 00:06:08,450 A_1 and

212 00:06:08,459 –> 00:06:09,769 here a set A_2

213 00:06:09,779 –> 00:06:10,470 .

214 00:06:10,899 –> 00:06:12,160 And let’s choose a split

215 00:06:12,170 –> 00:06:13,869 set A three here.

216 00:06:15,989 –> 00:06:17,980 And for these sets, we can

217 00:06:17,989 –> 00:06:19,589 now choose some

218 00:06:19,600 –> 00:06:21,100 constant c_i.

219 00:06:21,869 –> 00:06:23,100 Hence, we have for the graph

220 00:06:23,109 –> 00:06:24,869 of the function some value

221 00:06:24,880 –> 00:06:26,609 here over A_1,

222 00:06:26,619 –> 00:06:28,220 another value over A_2

223 00:06:28,230 –> 00:06:29,209 here.

224 00:06:29,640 –> 00:06:31,239 And maybe a lower one

225 00:06:31,250 –> 00:06:33,010 here for A_3

226 00:06:34,890 –> 00:06:36,820 and outside of this set, we

227 00:06:36,829 –> 00:06:38,089 are just at zero.

228 00:06:38,920 –> 00:06:40,420 If we now go back to our

229 00:06:40,429 –> 00:06:41,760 visualization of the

230 00:06:41,769 –> 00:06:43,420 integral, we know it’s given

231 00:06:43,429 –> 00:06:44,459 by rectangles.

232 00:06:44,470 –> 00:06:46,320 For example, here we have

233 00:06:46,329 –> 00:06:47,320 the rectangle

234 00:06:49,739 –> 00:06:51,549 with length given by a

235 00:06:51,559 –> 00:06:53,239 one and the height

236 00:06:53,250 –> 00:06:54,640 given by C one.

237 00:06:55,619 –> 00:06:57,420 Therefore, this rectangle

238 00:06:57,429 –> 00:06:58,549 here just

239 00:06:58,559 –> 00:07:00,160 represents the part of the

240 00:07:00,170 –> 00:07:01,959 integral that is given

241 00:07:01,970 –> 00:07:03,799 by C one times the

242 00:07:03,809 –> 00:07:05,540 measure of a one.

243 00:07:06,959 –> 00:07:08,820 And the contribution for

244 00:07:08,829 –> 00:07:10,359 the integral given by the

245 00:07:10,369 –> 00:07:12,100 set A two is just

246 00:07:12,109 –> 00:07:13,500 this rectangle here.

247 00:07:14,549 –> 00:07:16,130 And of course, now given

248 00:07:16,140 –> 00:07:17,609 by C two

249 00:07:17,649 –> 00:07:19,329 times the measure

250 00:07:19,339 –> 00:07:20,570 of A two.

251 00:07:21,269 –> 00:07:22,489 And then the last part is

252 00:07:22,500 –> 00:07:24,410 coming from a three which

253 00:07:24,420 –> 00:07:25,790 consists of three parts.

254 00:07:25,799 –> 00:07:27,779 But in the sum, of course,

255 00:07:27,790 –> 00:07:29,230 this is also C

256 00:07:29,239 –> 00:07:30,660 three times the

257 00:07:30,670 –> 00:07:32,149 measure of a

258 00:07:32,160 –> 00:07:32,709 three.

259 00:07:33,630 –> 00:07:35,010 And now you should see that

260 00:07:35,019 –> 00:07:36,790 we just have to add

261 00:07:36,799 –> 00:07:37,690 up all these

262 00:07:37,700 –> 00:07:39,500 contributions to get the

263 00:07:39,510 –> 00:07:40,929 full integral, to get the

264 00:07:40,940 –> 00:07:42,890 full area in this picture.

265 00:07:44,000 –> 00:07:45,540 Hence, again, for

266 00:07:45,549 –> 00:07:46,920 meaningful integral

267 00:07:46,929 –> 00:07:48,899 definition, we need a sum

268 00:07:48,910 –> 00:07:50,149 of all these parts.

269 00:07:50,160 –> 00:07:51,859 So I of

270 00:07:51,869 –> 00:07:53,700 F is then

271 00:07:53,709 –> 00:07:55,700 given by a sum

272 00:07:56,929 –> 00:07:58,869 and they have the same C

273 00:07:58,880 –> 00:08:00,549 I as before, but

274 00:08:00,559 –> 00:08:02,549 now not the characteristic

275 00:08:02,559 –> 00:08:03,589 function but the

276 00:08:03,600 –> 00:08:05,480 corresponding measures

277 00:08:05,640 –> 00:08:06,750 of the sets.

278 00:08:08,609 –> 00:08:10,510 And indeed, this will be

279 00:08:10,519 –> 00:08:12,380 our integral for

280 00:08:12,390 –> 00:08:14,070 simple functions in the end.

281 00:08:15,660 –> 00:08:17,399 However, at the moment,

282 00:08:17,519 –> 00:08:18,950 you should see immediately

283 00:08:18,959 –> 00:08:20,929 one problem with this definition.

284 00:08:22,059 –> 00:08:24,049 What happens if we have

285 00:08:24,059 –> 00:08:25,480 a set with volume

286 00:08:25,489 –> 00:08:26,480 infinity?

287 00:08:26,489 –> 00:08:28,040 So that this number

288 00:08:28,049 –> 00:08:29,640 here is the symbol

289 00:08:29,649 –> 00:08:30,559 infinity.

290 00:08:31,450 –> 00:08:33,429 There we could have for example,

291 00:08:34,090 –> 00:08:35,469 three as the C

292 00:08:35,760 –> 00:08:37,349 times infinity.

293 00:08:38,330 –> 00:08:39,549 Maybe that’s not a problem

294 00:08:39,558 –> 00:08:40,909 for you because it’s just

295 00:08:40,919 –> 00:08:42,460 infinity itself again.

296 00:08:42,739 –> 00:08:44,179 But then what happens if

297 00:08:44,190 –> 00:08:46,109 the next C here is

298 00:08:46,119 –> 00:08:48,070 minus two and it measures

299 00:08:48,080 –> 00:08:49,150 also infinity.

300 00:08:50,590 –> 00:08:51,979 And of course, this is not

301 00:08:51,989 –> 00:08:53,890 simply just one infinity.

302 00:08:53,900 –> 00:08:55,260 This is not defined at the

303 00:08:55,270 –> 00:08:55,770 moment.

304 00:08:57,070 –> 00:08:58,469 In fact, we have a

305 00:08:58,479 –> 00:09:00,229 problem with the definition

306 00:09:00,239 –> 00:09:00,859 in general.

307 00:09:02,169 –> 00:09:04,140 In order to solve this problem,

308 00:09:04,239 –> 00:09:05,539 there are essentially two

309 00:09:05,549 –> 00:09:06,770 ways one can go.

310 00:09:06,780 –> 00:09:08,510 Now on the one

311 00:09:08,520 –> 00:09:10,179 hand, one can restrict

312 00:09:10,190 –> 00:09:12,039 oneself to such simple

313 00:09:12,049 –> 00:09:13,979 functions where the corresponding

314 00:09:14,140 –> 00:09:16,020 AIS only have

315 00:09:16,030 –> 00:09:17,200 finite measure.

316 00:09:18,010 –> 00:09:19,559 Obviously, then there’s no

317 00:09:19,570 –> 00:09:21,429 problem with such infinities

318 00:09:21,440 –> 00:09:22,659 because there is no simple

319 00:09:22,669 –> 00:09:24,020 infinity involved in the

320 00:09:24,030 –> 00:09:24,750 whole sum.

321 00:09:25,570 –> 00:09:27,049 And on the other hand, one

322 00:09:27,059 –> 00:09:28,789 can demand that these

323 00:09:28,799 –> 00:09:30,179 constants have to be

324 00:09:30,190 –> 00:09:32,119 positive, then there

325 00:09:32,130 –> 00:09:33,650 might be infinity in the

326 00:09:33,659 –> 00:09:35,609 sun, but only with the same

327 00:09:35,619 –> 00:09:36,250 sign.

328 00:09:36,960 –> 00:09:38,280 And then of course, we don’t

329 00:09:38,289 –> 00:09:39,830 have a problem adding up

330 00:09:39,840 –> 00:09:41,380 positive infinities.

331 00:09:42,109 –> 00:09:44,030 And exactly this second

332 00:09:44,039 –> 00:09:45,890 possibility is the way

333 00:09:45,900 –> 00:09:47,159 I want to go here.

334 00:09:48,059 –> 00:09:49,960 Now, finally, we can put

335 00:09:49,969 –> 00:09:51,929 the integral into a

336 00:09:51,940 –> 00:09:52,799 definition

337 00:09:53,890 –> 00:09:54,520 for this.

338 00:09:54,530 –> 00:09:56,440 We will consider now, just

339 00:09:56,450 –> 00:09:58,299 functions defined on X.

340 00:09:59,070 –> 00:10:00,809 And as before, there should

341 00:10:00,820 –> 00:10:02,719 be step or simple

342 00:10:02,729 –> 00:10:03,409 functions.

343 00:10:04,140 –> 00:10:05,880 Moreover, we want now to

344 00:10:05,890 –> 00:10:07,859 consider only the positive

345 00:10:07,869 –> 00:10:09,659 ones or more concretely

346 00:10:09,669 –> 00:10:11,460 the non-negative ones.

347 00:10:12,599 –> 00:10:13,979 The whole set of such

348 00:10:13,989 –> 00:10:15,380 functions is now

349 00:10:15,390 –> 00:10:17,150 denoted by a

350 00:10:17,159 –> 00:10:19,059 curved S where I put

351 00:10:19,070 –> 00:10:20,460 in A plus

352 00:10:21,789 –> 00:10:23,609 and this S plus is now

353 00:10:23,619 –> 00:10:25,609 almost a vector space

354 00:10:25,950 –> 00:10:27,559 in the sense that we can

355 00:10:27,570 –> 00:10:29,090 add functions as we want.

356 00:10:29,349 –> 00:10:30,909 But we can’t scale them as

357 00:10:30,919 –> 00:10:32,369 we want because

358 00:10:32,380 –> 00:10:34,309 only positive scalars are

359 00:10:34,320 –> 00:10:35,799 allowed otherwise we would

360 00:10:35,809 –> 00:10:36,780 leave this space.

361 00:10:36,789 –> 00:10:38,570 So you should imagine

362 00:10:38,580 –> 00:10:40,169 this as a half space.

363 00:10:41,460 –> 00:10:43,179 If we go now back to our

364 00:10:43,190 –> 00:10:44,719 definition of simple

365 00:10:44,729 –> 00:10:45,859 functions, you

366 00:10:45,869 –> 00:10:47,599 immediately recognize

367 00:10:47,760 –> 00:10:49,239 that this representation

368 00:10:49,250 –> 00:10:50,880 here is not unique for the

369 00:10:50,890 –> 00:10:51,599 function F.

370 00:10:52,210 –> 00:10:54,030 For example, we could split

371 00:10:54,039 –> 00:10:55,760 the set A two into two

372 00:10:55,770 –> 00:10:56,429 parts.

373 00:10:56,659 –> 00:10:58,140 And then we have one

374 00:10:58,150 –> 00:11:00,119 term in addition in

375 00:11:00,130 –> 00:11:02,080 the sum here, but

376 00:11:02,090 –> 00:11:03,760 it does not change the function

377 00:11:03,770 –> 00:11:04,700 F itself.

378 00:11:04,710 –> 00:11:06,119 So the graph looks still

379 00:11:06,130 –> 00:11:07,799 the same just this

380 00:11:07,809 –> 00:11:09,119 sum is different.

381 00:11:10,030 –> 00:11:11,809 Therefore, another description

382 00:11:11,820 –> 00:11:13,710 which is independent of this

383 00:11:13,719 –> 00:11:15,309 representation for a simple

384 00:11:15,320 –> 00:11:16,809 function would be

385 00:11:18,229 –> 00:11:19,700 I have a measurable

386 00:11:19,710 –> 00:11:20,369 function.

387 00:11:20,440 –> 00:11:21,739 And it only takes

388 00:11:21,750 –> 00:11:23,700 finitely many values.

389 00:11:24,650 –> 00:11:26,549 In this case, I can always

390 00:11:26,559 –> 00:11:28,309 find such a representation

391 00:11:28,320 –> 00:11:28,630 here.

392 00:11:29,799 –> 00:11:30,780 The idea is now

393 00:11:30,789 –> 00:11:32,400 always choose a

394 00:11:32,409 –> 00:11:33,380 suitable

395 00:11:33,390 –> 00:11:34,969 representation for your

396 00:11:34,979 –> 00:11:36,140 simple function F.

397 00:11:36,950 –> 00:11:38,770 In our case here, if

398 00:11:38,780 –> 00:11:40,609 F comes from our S

399 00:11:40,619 –> 00:11:41,440 plus,

400 00:11:43,619 –> 00:11:45,479 then choose a representation

401 00:11:45,659 –> 00:11:46,640 as always.

402 00:11:46,650 –> 00:11:48,520 So we can write it as a sum.

403 00:11:49,340 –> 00:11:51,289 But now where the c_i’s

404 00:11:51,299 –> 00:11:52,770 are also non-

405 00:11:52,799 –> 00:11:53,559 negative.

406 00:11:55,109 –> 00:11:56,580 Of course, if the function

407 00:11:56,590 –> 00:11:58,049 is non-negative, then we

408 00:11:58,059 –> 00:11:59,669 can always choose such a

409 00:11:59,679 –> 00:12:00,570 representation.

410 00:12:01,880 –> 00:12:03,679 And now it’s very easy

411 00:12:03,690 –> 00:12:05,419 to define the Lebesgue integral.

412 00:12:06,429 –> 00:12:08,320 So this is our new notion

413 00:12:08,780 –> 00:12:10,570 and of course, often we will

414 00:12:10,580 –> 00:12:12,419 ignore the Lebesgue and

415 00:12:12,429 –> 00:12:13,710 just speak of the

416 00:12:13,719 –> 00:12:15,530 integral because this is

417 00:12:15,539 –> 00:12:17,530 the integral we define here.

418 00:12:18,419 –> 00:12:19,789 And to be more concrete,

419 00:12:19,799 –> 00:12:21,190 we also would say with

420 00:12:21,200 –> 00:12:22,799 respect to our

421 00:12:22,809 –> 00:12:24,409 given measure mu.

422 00:12:25,809 –> 00:12:27,349 And this integral is now

423 00:12:27,359 –> 00:12:29,070 given as we already

424 00:12:29,080 –> 00:12:29,510 know.

425 00:12:29,520 –> 00:12:31,390 And also denoted by

426 00:12:31,400 –> 00:12:33,340 I of F where we

427 00:12:33,349 –> 00:12:34,859 just add up

428 00:12:35,150 –> 00:12:36,780 the measures of the

429 00:12:36,789 –> 00:12:38,539 sets A I where we

430 00:12:38,549 –> 00:12:40,530 scale them also by the

431 00:12:40,539 –> 00:12:41,169 CIS.

432 00:12:42,039 –> 00:12:43,260 So we have C I

433 00:12:43,270 –> 00:12:44,900 times the measure

434 00:12:44,909 –> 00:12:46,369 of A I.

435 00:12:47,710 –> 00:12:49,590 And because the C I are

436 00:12:49,599 –> 00:12:51,409 non-negative and the measure

437 00:12:51,419 –> 00:12:53,090 itself is non-negative or

438 00:12:53,099 –> 00:12:54,260 in the worst case just

439 00:12:54,270 –> 00:12:56,030 infinity, we know this

440 00:12:56,039 –> 00:12:58,020 integral is now also in the

441 00:12:58,070 –> 00:12:59,729 interval zero

442 00:12:59,840 –> 00:13:01,390 to infinity where we

443 00:13:01,400 –> 00:13:03,090 include both parts.

444 00:13:04,219 –> 00:13:05,479 One fact I can give you

445 00:13:05,489 –> 00:13:07,299 immediately this definition

446 00:13:07,309 –> 00:13:08,929 for integral is a

447 00:13:08,940 –> 00:13:10,400 well-defined

448 00:13:10,510 –> 00:13:11,400 object.

449 00:13:13,080 –> 00:13:14,940 This means that it does not

450 00:13:14,950 –> 00:13:16,520 depend on the chosen

451 00:13:16,530 –> 00:13:18,320 representation for our simple

452 00:13:18,330 –> 00:13:19,380 function F.

453 00:13:19,619 –> 00:13:20,859 So if you choose another

454 00:13:20,869 –> 00:13:22,260 representation with this

455 00:13:22,270 –> 00:13:24,260 property, you get the same

456 00:13:24,270 –> 00:13:26,099 number out, maybe

457 00:13:26,109 –> 00:13:27,489 this is a thing you could

458 00:13:27,500 –> 00:13:28,580 try to prove.

459 00:13:29,409 –> 00:13:31,070 However, of course, the

460 00:13:31,080 –> 00:13:32,369 visualization should be the

461 00:13:32,380 –> 00:13:33,369 important thing.

462 00:13:33,409 –> 00:13:35,280 So it does not matter how

463 00:13:35,289 –> 00:13:36,750 you split up these

464 00:13:36,760 –> 00:13:37,919 rectangles here.

465 00:13:38,090 –> 00:13:39,710 Yeah, the sum or the sum

466 00:13:39,719 –> 00:13:41,630 of the areas should be always

467 00:13:41,640 –> 00:13:43,330 the same no

468 00:13:43,340 –> 00:13:44,929 matter how you split up the

469 00:13:44,940 –> 00:13:46,809 set here on the X axis or

470 00:13:46,820 –> 00:13:48,659 you split up the C

471 00:13:48,669 –> 00:13:50,359 values on the Y axis.

472 00:13:51,349 –> 00:13:52,700 In other words, it makes

473 00:13:52,710 –> 00:13:54,270 sense that this integral

474 00:13:54,280 –> 00:13:55,760 here is indeed well-

475 00:13:55,770 –> 00:13:56,369 defined.

476 00:13:57,380 –> 00:13:59,359 And finally, I can give you

477 00:13:59,369 –> 00:14:01,099 the usual notation one

478 00:14:01,109 –> 00:14:02,419 uses for the integral,

479 00:14:03,070 –> 00:14:04,900 of course, an integral sign,

480 00:14:04,919 –> 00:14:06,640 we put the set X

481 00:14:06,650 –> 00:14:08,280 here then comes the

482 00:14:08,289 –> 00:14:10,010 function F and

483 00:14:10,020 –> 00:14:11,530 then the measure mu

484 00:14:11,539 –> 00:14:13,380 by d

485 00:14:13,539 –> 00:14:13,929 mu

486 00:14:15,130 –> 00:14:16,530 sometimes also a

487 00:14:16,539 –> 00:14:17,950 variable is needed.

488 00:14:17,960 –> 00:14:19,650 And the notation looks almost

489 00:14:19,659 –> 00:14:20,349 the same.

490 00:14:20,859 –> 00:14:21,830 So you just

491 00:14:21,950 –> 00:14:23,549 include a variable

492 00:14:23,559 –> 00:14:24,950 name for the function.

493 00:14:24,960 –> 00:14:26,440 So a lower case X

494 00:14:26,450 –> 00:14:28,030 here and then d mu

495 00:14:28,840 –> 00:14:29,619 X.

496 00:14:31,030 –> 00:14:32,890 So this is the Lebesgue integral

497 00:14:32,929 –> 00:14:34,729 for step or

498 00:14:34,739 –> 00:14:35,890 simple functions.

499 00:14:36,640 –> 00:14:38,219 The idea is now that we

500 00:14:38,229 –> 00:14:40,059 expand this definition

501 00:14:40,070 –> 00:14:41,489 such that we can also

502 00:14:41,500 –> 00:14:43,159 integrate more complicated

503 00:14:43,169 –> 00:14:43,809 functions.

504 00:14:44,659 –> 00:14:46,340 But before we do that, let’s

505 00:14:46,349 –> 00:14:48,320 first look at some properties

506 00:14:48,330 –> 00:14:49,719 this integral has

507 00:14:50,419 –> 00:14:51,919 firstly, it is

508 00:14:51,929 –> 00:14:53,489 almost linear.

509 00:14:53,960 –> 00:14:55,590 I already told you

510 00:14:55,619 –> 00:14:57,130 we almost have vector

511 00:14:57,140 –> 00:14:59,030 space for our S plus.

512 00:14:59,179 –> 00:15:00,630 And in this sense, it’s

513 00:15:00,640 –> 00:15:02,450 linear, we

514 00:15:02,460 –> 00:15:03,900 just have to restrict the

515 00:15:03,909 –> 00:15:05,890 scalars to non-negative

516 00:15:05,900 –> 00:15:06,679 numbers.

517 00:15:07,440 –> 00:15:08,780 And then we can pull out

518 00:15:08,789 –> 00:15:09,900 the addition sign.

519 00:15:09,909 –> 00:15:11,809 And also the scalars,

520 00:15:12,309 –> 00:15:14,250 hence this equality

521 00:15:14,260 –> 00:15:15,809 holds for all simple

522 00:15:15,820 –> 00:15:17,640 functions FG and

523 00:15:17,650 –> 00:15:19,010 for all scalars

524 00:15:19,179 –> 00:15:21,070 alpha beta greater

525 00:15:21,080 –> 00:15:22,299 or equal to zero.

526 00:15:23,359 –> 00:15:25,010 And of course, this immediately

527 00:15:25,020 –> 00:15:26,820 comes from this sum here.

528 00:15:28,119 –> 00:15:29,979 Also by using a, we

529 00:15:29,989 –> 00:15:31,780 can now show that if we

530 00:15:31,789 –> 00:15:33,070 have a step function or a

531 00:15:33,080 –> 00:15:34,940 simple function that is always

532 00:15:34,950 –> 00:15:36,219 bigger than another one,

533 00:15:36,229 –> 00:15:37,989 so we have F less or

534 00:15:38,000 –> 00:15:39,140 equal than G,

535 00:15:39,799 –> 00:15:41,419 then also the area

536 00:15:41,530 –> 00:15:43,280 between the graph

537 00:15:43,289 –> 00:15:44,780 and the X axis should be

538 00:15:44,789 –> 00:15:46,619 bigger for G than for F.

539 00:15:47,150 –> 00:15:48,349 And of course, this is what

540 00:15:48,359 –> 00:15:49,359 the integral tells us.

541 00:15:49,369 –> 00:15:51,280 So I of F

542 00:15:51,289 –> 00:15:53,179 is also less or equal

543 00:15:53,190 –> 00:15:54,960 than I of G.

544 00:15:56,020 –> 00:15:57,599 And this is what one calls

545 00:15:57,609 –> 00:15:58,710 monotonicity.

546 00:15:59,659 –> 00:16:01,270 In fact, this monotonic

547 00:16:01,280 –> 00:16:03,109 behavior is what we now

548 00:16:03,119 –> 00:16:04,989 can use to expand our

549 00:16:05,000 –> 00:16:06,729 definition to

550 00:16:06,739 –> 00:16:08,729 general measurable maps

551 00:16:09,390 –> 00:16:10,250 in order to get a

552 00:16:10,260 –> 00:16:12,039 visualization, let’s

553 00:16:12,049 –> 00:16:13,919 sketch again a graph.

554 00:16:14,739 –> 00:16:16,619 Now this is the graph of

555 00:16:16,630 –> 00:16:18,119 some measurable map

556 00:16:18,349 –> 00:16:20,049 which is not a simple function.

557 00:16:20,059 –> 00:16:20,840 As you can see.

558 00:16:21,719 –> 00:16:23,299 Now, the integral should

559 00:16:23,309 –> 00:16:25,130 be now again represented

560 00:16:25,140 –> 00:16:26,940 by the area below

561 00:16:26,950 –> 00:16:28,880 the graph and the X axis.

562 00:16:29,799 –> 00:16:31,429 However, we only have this

563 00:16:31,440 –> 00:16:33,190 notion for the moment for

564 00:16:33,200 –> 00:16:34,469 the simple functions.

565 00:16:35,270 –> 00:16:36,390 And as you know, by now,

566 00:16:36,400 –> 00:16:38,090 a simple function has

567 00:16:38,099 –> 00:16:39,469 only finitely many

568 00:16:39,479 –> 00:16:40,260 values.

569 00:16:40,679 –> 00:16:41,770 And this blue curve here

570 00:16:41,780 –> 00:16:43,650 shows you there are infinitely

571 00:16:43,659 –> 00:16:44,770 many values.

572 00:16:45,539 –> 00:16:46,929 The whole idea of the Lebesgue

573 00:16:47,080 –> 00:16:48,770 integration is now to

574 00:16:48,780 –> 00:16:50,440 approximate the function

575 00:16:50,450 –> 00:16:51,830 by finitely many

576 00:16:51,840 –> 00:16:52,469 values.

577 00:16:53,549 –> 00:16:55,530 So I just choose some values

578 00:16:55,539 –> 00:16:57,010 here on the y axis

579 00:16:57,020 –> 00:16:58,549 such that the whole

580 00:16:58,869 –> 00:17:00,320 curve is in some sense

581 00:17:00,330 –> 00:17:01,049 covered.

582 00:17:02,390 –> 00:17:04,020 And now the idea is to

583 00:17:04,030 –> 00:17:05,800 define a suitable

584 00:17:05,810 –> 00:17:06,848 simple function.

585 00:17:08,218 –> 00:17:09,608 Therefore, maybe I choose

586 00:17:09,618 –> 00:17:11,159 this interval here

587 00:17:11,568 –> 00:17:13,358 and look what happens to

588 00:17:13,368 –> 00:17:14,968 the values of the function

589 00:17:14,979 –> 00:17:15,588 there.

590 00:17:15,878 –> 00:17:17,458 And now you see we have two

591 00:17:17,468 –> 00:17:19,358 parts that are

592 00:17:19,368 –> 00:17:21,138 mapped to this

593 00:17:21,148 –> 00:17:22,038 interval here.

594 00:17:22,939 –> 00:17:24,550 For the X axis, this

595 00:17:24,560 –> 00:17:25,489 looks like this.

596 00:17:25,500 –> 00:17:27,390 So we have this part here

597 00:17:27,400 –> 00:17:29,209 that ends here

598 00:17:29,520 –> 00:17:31,290 and we have this part that

599 00:17:31,300 –> 00:17:32,560 starts here

600 00:17:32,949 –> 00:17:34,780 and also ends

601 00:17:34,790 –> 00:17:35,209 here.

602 00:17:36,109 –> 00:17:38,050 This means we have one part

603 00:17:38,060 –> 00:17:39,979 of our set here and the other

604 00:17:39,989 –> 00:17:41,530 part here.

605 00:17:42,810 –> 00:17:44,170 And of course, this should

606 00:17:44,180 –> 00:17:45,800 be our set that we

607 00:17:45,810 –> 00:17:47,430 usually call A

608 00:17:47,619 –> 00:17:48,270 I.

609 00:17:48,280 –> 00:17:49,729 So this is just one A

610 00:17:49,739 –> 00:17:50,810 I and the

611 00:17:50,819 –> 00:17:52,479 corresponding C

612 00:17:52,619 –> 00:17:54,349 you find now here.

613 00:17:54,500 –> 00:17:56,459 So this would be our C I.

614 00:17:57,859 –> 00:17:59,750 So I chose the lower part

615 00:17:59,760 –> 00:18:01,199 of the interval because then

616 00:18:01,209 –> 00:18:03,199 our step function is also

617 00:18:03,260 –> 00:18:05,239 below the graph of the

618 00:18:05,250 –> 00:18:05,839 function.

619 00:18:05,849 –> 00:18:07,780 So this would be the

620 00:18:07,790 –> 00:18:09,040 value of our step function

621 00:18:09,050 –> 00:18:09,319 here.

622 00:18:10,410 –> 00:18:12,339 And now you see all the other

623 00:18:12,349 –> 00:18:14,040 values on the Y axis

624 00:18:14,050 –> 00:18:15,810 give us the other

625 00:18:15,819 –> 00:18:17,550 c_i’s and also the

626 00:18:17,560 –> 00:18:19,180 corresponding A_i’s

627 00:18:20,060 –> 00:18:21,880 therefore, with this decomposition

628 00:18:21,890 –> 00:18:23,640 of the Y axis, we get

629 00:18:23,650 –> 00:18:25,540 out the usual step function

630 00:18:25,550 –> 00:18:27,489 where we have our C is there.

631 00:18:27,569 –> 00:18:29,359 And the so defined A

632 00:18:29,369 –> 00:18:31,300 is where we have now the

633 00:18:31,310 –> 00:18:32,800 characteristic function here.

634 00:18:33,670 –> 00:18:35,359 So this is a new simple function

635 00:18:35,369 –> 00:18:36,630 which I should call

636 00:18:36,859 –> 00:18:37,420 h now.

637 00:18:38,310 –> 00:18:39,880 And the important part is

638 00:18:39,890 –> 00:18:41,729 that h itself

639 00:18:41,739 –> 00:18:43,010 lies always

640 00:18:43,020 –> 00:18:44,989 below F so

641 00:18:45,000 –> 00:18:46,609 always below the graph that

642 00:18:46,619 –> 00:18:47,479 is blue here.

643 00:18:48,729 –> 00:18:50,229 Therefore, if we want to

644 00:18:50,239 –> 00:18:52,209 contain the monotonicity

645 00:18:52,219 –> 00:18:53,619 in the general integral,

646 00:18:53,630 –> 00:18:55,390 we now have an estimate

647 00:18:55,400 –> 00:18:57,329 for the real integral value

648 00:18:57,339 –> 00:18:58,989 of the function F.

649 00:18:59,010 –> 00:19:00,160 We know that the integral

650 00:19:00,170 –> 00:19:01,709 of this step function is

651 00:19:01,719 –> 00:19:03,619 smaller than the real

652 00:19:03,630 –> 00:19:04,349 integral.

653 00:19:05,160 –> 00:19:07,020 And this gives now rise to

654 00:19:07,030 –> 00:19:08,380 the following definition.

655 00:19:09,319 –> 00:19:11,150 Hence, we choose a positive

656 00:19:11,160 –> 00:19:13,140 or better a non-negative

657 00:19:13,150 –> 00:19:14,500 function F that is

658 00:19:14,510 –> 00:19:16,099 defined on our measure

659 00:19:16,109 –> 00:19:17,060 space X.

660 00:19:17,780 –> 00:19:18,959 And of course, it should

661 00:19:18,969 –> 00:19:20,359 be measurable.

662 00:19:21,420 –> 00:19:23,260 And now for each decomposition

663 00:19:23,270 –> 00:19:25,060 of the Y axis, we can

664 00:19:25,069 –> 00:19:26,260 choose such a

665 00:19:26,270 –> 00:19:28,020 step function or simple

666 00:19:28,030 –> 00:19:29,920 function h that lies

667 00:19:29,930 –> 00:19:31,800 pointwisely below

668 00:19:31,810 –> 00:19:32,930 the graph of the function

669 00:19:32,939 –> 00:19:33,829 F itself.

670 00:19:34,650 –> 00:19:36,500 And if we choose the function

671 00:19:36,510 –> 00:19:38,369 h out of the

672 00:19:38,510 –> 00:19:39,760 step function that are

673 00:19:39,770 –> 00:19:41,660 positive or non negative.

674 00:19:41,670 –> 00:19:43,589 So S plus, we know

675 00:19:43,599 –> 00:19:44,790 we can look at the

676 00:19:44,800 –> 00:19:46,500 integral of this

677 00:19:46,510 –> 00:19:47,030 h.

678 00:19:48,040 –> 00:19:49,829 What we have is then a whole

679 00:19:49,839 –> 00:19:51,400 set of integral

680 00:19:51,410 –> 00:19:53,280 values where the only condition

681 00:19:53,290 –> 00:19:55,180 is that our step function

682 00:19:55,189 –> 00:19:57,089 is always below

683 00:19:57,189 –> 00:19:58,810 the real function F.

684 00:20:01,239 –> 00:20:02,900 The general idea is now

685 00:20:02,910 –> 00:20:03,599 OK.

686 00:20:03,609 –> 00:20:05,500 So this integral value for

687 00:20:05,510 –> 00:20:07,459 the step function is always

688 00:20:07,469 –> 00:20:08,939 smaller than the

689 00:20:08,949 –> 00:20:10,859 actual integral value for

690 00:20:10,869 –> 00:20:12,150 the function F.

691 00:20:12,160 –> 00:20:13,609 And we should get closer

692 00:20:13,619 –> 00:20:15,160 and closer to this value

693 00:20:15,170 –> 00:20:17,040 if we choose a finer and

694 00:20:17,050 –> 00:20:18,890 finer decomposition of the

695 00:20:18,900 –> 00:20:20,800 Y axis, so

696 00:20:20,810 –> 00:20:22,359 we approximate something

697 00:20:22,369 –> 00:20:23,670 with the set here.

698 00:20:23,680 –> 00:20:25,040 And of course, this should

699 00:20:25,050 –> 00:20:26,859 be the supremum of the

700 00:20:26,869 –> 00:20:27,319 set.

701 00:20:28,300 –> 00:20:29,579 In fact, this is what we

702 00:20:29,589 –> 00:20:31,069 choose as the

703 00:20:31,079 –> 00:20:32,430 definition of the

704 00:20:32,439 –> 00:20:34,209 integral for our function

705 00:20:34,219 –> 00:20:34,800 F.

706 00:20:34,810 –> 00:20:36,560 So we have F d mu

707 00:20:37,099 –> 00:20:39,069 and also this is our measure

708 00:20:39,079 –> 00:20:39,900 space X.

709 00:20:40,750 –> 00:20:42,540 So the integral of a nonne

710 00:20:43,199 –> 00:20:45,089 measurable map

711 00:20:45,099 –> 00:20:47,089 F is given

712 00:20:47,099 –> 00:20:48,859 by the supremum of

713 00:20:48,869 –> 00:20:50,589 all integral values for

714 00:20:50,599 –> 00:20:52,130 step functions that

715 00:20:52,140 –> 00:20:53,989 lie below the function

716 00:20:54,000 –> 00:20:54,479 F.

717 00:20:55,390 –> 00:20:56,609 And that is now the

718 00:20:56,619 –> 00:20:58,119 definition of the Lebesgue

719 00:20:58,439 –> 00:20:59,089 integral.

720 00:20:59,770 –> 00:21:01,579 And we also see this is well

721 00:21:01,589 –> 00:21:02,910 defined the

722 00:21:02,920 –> 00:21:04,650 supremum of a

723 00:21:04,660 –> 00:21:06,550 set in the real numbers always

724 00:21:06,560 –> 00:21:07,390 exists.

725 00:21:07,619 –> 00:21:09,060 In the worst case, it would

726 00:21:09,069 –> 00:21:10,030 be infinity.

727 00:21:11,010 –> 00:21:12,400 Therefore, in addition, we

728 00:21:12,410 –> 00:21:14,300 also have another definition

729 00:21:14,800 –> 00:21:16,599 F is called mu-integrable

730 00:21:19,239 –> 00:21:20,780 if the integral is

731 00:21:20,790 –> 00:21:21,619 finite.

732 00:21:22,920 –> 00:21:24,739 So it’s not the symbol

733 00:21:24,750 –> 00:21:25,699 infinity.

734 00:21:27,770 –> 00:21:29,219 And there we have it, this

735 00:21:29,229 –> 00:21:30,900 is our result for today,

736 00:21:30,910 –> 00:21:32,530 the Lebesgue integral in

737 00:21:32,540 –> 00:21:34,219 complete generality

738 00:21:34,280 –> 00:21:35,750 because you see the only

739 00:21:35,760 –> 00:21:37,420 thing we needed here

740 00:21:37,550 –> 00:21:38,969 was a measure space

741 00:21:38,979 –> 00:21:39,599 X.

742 00:21:40,010 –> 00:21:41,430 So we only needed to know

743 00:21:41,439 –> 00:21:43,030 how to measure the

744 00:21:43,040 –> 00:21:45,030 volumes of these sets

745 00:21:45,040 –> 00:21:46,489 on the X axis.

746 00:21:47,160 –> 00:21:48,689 However, what we use is that

747 00:21:48,699 –> 00:21:50,339 we map into the real

748 00:21:50,349 –> 00:21:51,069 numbers.

749 00:21:51,079 –> 00:21:52,910 So indeed, here we have R

750 00:21:52,920 –> 00:21:54,250 on the Y axis.

751 00:21:54,300 –> 00:21:55,770 Therefore, we can do this

752 00:21:55,780 –> 00:21:57,380 decomposition, we can’t do

753 00:21:57,390 –> 00:21:58,310 this decomposition on the

754 00:21:58,319 –> 00:22:00,150 X axis because there’s no

755 00:22:00,160 –> 00:22:02,089 order is nothing more than

756 00:22:02,099 –> 00:22:03,630 just the measure.

757 00:22:03,880 –> 00:22:05,469 But on the y axis, we can

758 00:22:05,479 –> 00:22:05,869 do it.

759 00:22:06,949 –> 00:22:08,119 Now, we should always keep

760 00:22:08,130 –> 00:22:09,920 in mind the Lebesgue integral

761 00:22:09,930 –> 00:22:11,849 is just defined within

762 00:22:11,859 –> 00:22:13,689 supremum where we use

763 00:22:13,699 –> 00:22:15,369 simple functions to

764 00:22:15,380 –> 00:22:17,010 define the integral first.

765 00:22:18,079 –> 00:22:19,699 So this was a long video

766 00:22:19,709 –> 00:22:21,569 today and I hope you

767 00:22:21,579 –> 00:22:22,699 learned something here.

768 00:22:23,530 –> 00:22:25,270 Of course, we had to define

769 00:22:25,280 –> 00:22:26,290 a lot of stuff.

770 00:22:26,380 –> 00:22:28,270 But in the end, you saw

771 00:22:28,599 –> 00:22:30,400 it is just a simple function

772 00:22:30,410 –> 00:22:31,670 where we can write down the

773 00:22:31,680 –> 00:22:33,020 integral immediately.

774 00:22:33,030 –> 00:22:34,540 And then the

775 00:22:34,550 –> 00:22:36,469 general integral is just

776 00:22:36,479 –> 00:22:38,150 an approximation concept.

777 00:22:39,010 –> 00:22:40,540 In the next video, I will

778 00:22:40,550 –> 00:22:41,790 continue with the Lebesgue

779 00:22:41,800 –> 00:22:43,709 integral and also

780 00:22:43,719 –> 00:22:45,079 write down a lot of

781 00:22:45,089 –> 00:22:46,939 properties we have for it.

782 00:22:47,780 –> 00:22:49,589 And afterwards, you will

783 00:22:49,599 –> 00:22:51,489 finally see why the Lebesgue

784 00:22:51,609 –> 00:22:53,560 integral is so powerful,

785 00:22:54,619 –> 00:22:56,180 then have a nice day

786 00:22:56,250 –> 00:22:57,890 and see you next time.

787 00:22:58,310 –> 00:22:58,920 Bye.

Q1: Let $(X, \mathcal{A}, \mu)$ be a measure space and $\chi_A: X \rightarrow \mathbb{R}$ the indicator function for $A \in \mathcal{A}$. What is the integral of $\chi_A$ w.r.t. $\mu$ (by definition)?

A1: $I(\chi_A) = \mu(A^c)$

A2: $I(\chi_A) = 1-\mu(A)$

A3: $I(\chi_A) = \mu(A)$

A4: $I(\chi_A) = \mu(A^c) + \mu(A)$

Q2: Let $(X, \mathcal{A}, \mu)$ be a measure space. Which property does a simple function $f: X \rightarrow \mathbb{R}$ not fulfil?

A1: ${ f(x) \mid x \in X }$ is finite

A2: $f$ is measurable

A3: $f(x) \in \mathbb{R}$ for all $x \in X$

A4: $f^{-1}({\mathbb{R}}) = X$

A5: ${ f(x) \mid x \in X } = \mathbb{R}$

Q3: Consider $(X, \mathcal{A}) = (\mathbb{R}, \mathcal{B}(\mathbb{R}) )$. Is the following function $f: \mathbb{R} \rightarrow \mathbb{R}$ a simple function? $$ f(x) := \sum_{i=1}^{100} i \cdot \chi_{[0,i]}(x) $$

A1: Yes.

A2: No.

Q4: Consider $(X, \mathcal{A}) = (\mathbb{R}, \mathcal{B}(\mathbb{R}) )$. Is the following function $f: \mathbb{R} \rightarrow \mathbb{R}$ a simple function? $$ f(x) := \chi_{[0,1]}(x) - \chi_{\mathbb{R}}(x) $$

A1: Yes, it is.

A2: No, it isn’t.

Q5: Let $(X, \mathcal{A}, \mu)$ be a measure space. Which of the following statements is not a property of the Lebesgue integral (w.r.t. $\mu$) of non-negative simple functions as a map $I: \mathcal{S}^+ \rightarrow [0,\infty]$.

A1: $I(f+g) = I(f) + I(g)$.

A2: $I(\alpha f) = \alpha I(f)$ for $\alpha \geq 0$.

A3: $f < g ~~\Rightarrow ~~ I(f) < I(g)$

Q6: Let $(X, \mathcal{A}, \mu)$ be a measure space. Can we define the Lebesgue integral for every measurable function $f: X \rightarrow [0,\infty)$?

A1: Yes and we have $\int_X f , d \mu \in \mathbb{R}$.

A2: Yes and we have $\int_X f , d \mu \in \mathbb{C}$.

A3: Yes and we have $\int_X f , d \mu \in [0,\infty]$.

A4: No, there are exceptions.