1 00:00:00,479 –> 00:00:02,269 Hello and welcome to a

2 00:00:02,279 –> 00:00:03,900 new video in the topic of

3 00:00:03,910 –> 00:00:05,010 measure theory.

4 00:00:05,710 –> 00:00:07,409 First, let me thank all the

5 00:00:07,420 –> 00:00:09,170 nice people that support

6 00:00:09,180 –> 00:00:10,890 this channel on steady.

7 00:00:11,789 –> 00:00:13,649 And today’s topic is Lebesgue-Stieltjes

8 00:00:14,579 –> 00:00:15,329 measures.

9 00:00:16,110 –> 00:00:17,389 It sounds like something

10 00:00:17,399 –> 00:00:19,090 different than a normal measure,

11 00:00:19,399 –> 00:00:21,110 but it’s not, it’s just a

12 00:00:21,120 –> 00:00:22,840 way to construct an

13 00:00:22,850 –> 00:00:23,969 ordinary measure.

14 00:00:24,899 –> 00:00:26,829 And in this way, we can start,

15 00:00:26,840 –> 00:00:28,450 we will now construct a

16 00:00:28,459 –> 00:00:29,020 measure.

17 00:00:29,809 –> 00:00:31,569 The only thing we need here

18 00:00:31,579 –> 00:00:33,069 is a monotonically

19 00:00:33,080 –> 00:00:34,590 increasing function

20 00:00:34,599 –> 00:00:36,549 defined on the real number

21 00:00:36,560 –> 00:00:37,020 line.

22 00:00:37,360 –> 00:00:38,979 And when I say monotonically

23 00:00:38,990 –> 00:00:40,790 increasing, I mean

24 00:00:40,799 –> 00:00:41,990 non-decreasing.

25 00:00:42,880 –> 00:00:44,580 In other words, a constant

26 00:00:44,590 –> 00:00:46,150 function is also

27 00:00:46,159 –> 00:00:47,889 monotonically increasing.

28 00:00:48,819 –> 00:00:50,159 The best way to see that

29 00:00:50,169 –> 00:00:51,990 is of course to draw a short

30 00:00:52,009 –> 00:00:53,240 graph of such a function.

31 00:00:54,250 –> 00:00:55,500 So maybe the function is

32 00:00:55,509 –> 00:00:57,200 first increasing here

33 00:00:58,009 –> 00:00:59,880 then at it is constant

34 00:01:01,849 –> 00:01:03,259 and maybe, then we have a

35 00:01:03,270 –> 00:01:05,029 jump point at this point.

36 00:01:05,650 –> 00:01:07,419 So no value here, the value

37 00:01:07,430 –> 00:01:09,150 is here then still the

38 00:01:09,160 –> 00:01:09,989 constant

39 00:01:11,059 –> 00:01:12,550 and maybe after this, it’s

40 00:01:12,559 –> 00:01:13,910 still increasing and then

41 00:01:13,919 –> 00:01:15,839 comes the next jump point.

42 00:01:17,360 –> 00:01:19,300 So here’s the point and then

43 00:01:19,309 –> 00:01:20,910 it goes to an

44 00:01:20,919 –> 00:01:22,290 increasing function till

45 00:01:22,300 –> 00:01:24,029 here and then maybe it’s

46 00:01:24,040 –> 00:01:25,309 constant after that.

47 00:01:26,750 –> 00:01:27,209 OK.

48 00:01:27,220 –> 00:01:29,129 So this is a typical example

49 00:01:29,139 –> 00:01:30,290 of a graph of a

50 00:01:30,300 –> 00:01:31,930 monotonically increasing

51 00:01:31,940 –> 00:01:32,459 function.

52 00:01:33,459 –> 00:01:35,379 In particular, it’s allowed

53 00:01:35,389 –> 00:01:36,660 that the function is

54 00:01:36,669 –> 00:01:38,279 constant at some

55 00:01:38,290 –> 00:01:38,919 parts.

56 00:01:39,610 –> 00:01:41,279 And there could be also points

57 00:01:41,290 –> 00:01:42,470 where the function is non

58 00:01:42,569 –> 00:01:43,459 continuous.

59 00:01:43,470 –> 00:01:45,099 So where we have jumps,

60 00:01:45,529 –> 00:01:46,910 but of course, the jumps

61 00:01:46,919 –> 00:01:48,540 has to go upwards.

62 00:01:49,769 –> 00:01:51,430 Now, with the help of such

63 00:01:51,440 –> 00:01:53,069 a function, we want to

64 00:01:53,080 –> 00:01:54,699 measure the length of

65 00:01:54,709 –> 00:01:55,510 intervals.

66 00:01:56,099 –> 00:01:57,629 First, I want to calculate

67 00:01:57,639 –> 00:01:59,470 intervals of the form A

68 00:01:59,480 –> 00:02:01,389 B where A is

69 00:02:01,400 –> 00:02:02,989 included but B is

70 00:02:03,000 –> 00:02:03,870 excluded

71 00:02:04,900 –> 00:02:06,510 for better visualization,

72 00:02:06,519 –> 00:02:08,070 I can put that here in the

73 00:02:08,080 –> 00:02:08,690 picture.

74 00:02:08,699 –> 00:02:10,589 So here’s A and

75 00:02:10,600 –> 00:02:11,809 there’s our B,

76 00:02:13,630 –> 00:02:15,559 obviously, the normal notion

77 00:02:15,570 –> 00:02:17,130 of the length would be just

78 00:02:17,139 –> 00:02:18,850 B minus

79 00:02:18,860 –> 00:02:19,309 A.

80 00:02:20,259 –> 00:02:22,119 But here I want that F

81 00:02:22,130 –> 00:02:24,039 as a function scales

82 00:02:24,050 –> 00:02:25,720 the length of this interval,

83 00:02:26,139 –> 00:02:27,639 this means that the length

84 00:02:27,649 –> 00:02:29,619 should be longer where the

85 00:02:29,630 –> 00:02:30,929 increase of the function

86 00:02:30,940 –> 00:02:32,179 is stronger.

87 00:02:33,250 –> 00:02:34,669 Therefore, we have to look

88 00:02:34,679 –> 00:02:36,449 what the function does with

89 00:02:36,460 –> 00:02:37,190 the interval.

90 00:02:37,300 –> 00:02:39,110 So we look at the

91 00:02:39,119 –> 00:02:40,729 image of the

92 00:02:40,740 –> 00:02:42,160 interval under the

93 00:02:42,169 –> 00:02:43,229 function F

94 00:02:44,220 –> 00:02:44,940 in the picture.

95 00:02:44,949 –> 00:02:46,419 This would roughly look like

96 00:02:46,429 –> 00:02:46,669 this.

97 00:02:46,679 –> 00:02:48,020 So we have a point

98 00:02:48,229 –> 00:02:50,100 FB here

99 00:02:50,839 –> 00:02:52,320 and F of A

100 00:02:52,330 –> 00:02:53,029 here

101 00:02:55,110 –> 00:02:56,839 scaling the length of this

102 00:02:56,850 –> 00:02:58,009 interval of the function

103 00:02:58,020 –> 00:02:58,419 F.

104 00:02:58,429 –> 00:03:00,410 Now just means we look at

105 00:03:00,419 –> 00:03:01,830 the Y axis here,

106 00:03:02,250 –> 00:03:04,000 which means the length of

107 00:03:04,009 –> 00:03:05,490 this interval which is

108 00:03:05,500 –> 00:03:07,220 just F of

109 00:03:07,229 –> 00:03:08,850 B minus

110 00:03:08,919 –> 00:03:10,110 F of A.

111 00:03:12,059 –> 00:03:12,630 OK.

112 00:03:12,839 –> 00:03:14,789 Now, I hope you immediately

113 00:03:14,800 –> 00:03:16,789 recognized a mistake here.

114 00:03:17,600 –> 00:03:19,589 You see B is not

115 00:03:19,600 –> 00:03:21,259 a point in our interval.

116 00:03:22,050 –> 00:03:23,479 Therefore, it would be wrong

117 00:03:23,490 –> 00:03:24,919 to use this point.

118 00:03:24,929 –> 00:03:26,729 So the image of B we

119 00:03:26,740 –> 00:03:28,559 should rather use this point

120 00:03:28,570 –> 00:03:29,179 here.

121 00:03:29,460 –> 00:03:31,389 This one is the correct point

122 00:03:31,399 –> 00:03:32,710 that describes the right

123 00:03:32,720 –> 00:03:34,300 hand bound of the

124 00:03:34,309 –> 00:03:35,539 image of A B.

125 00:03:36,479 –> 00:03:38,039 Therefore, on the Y axis,

126 00:03:38,050 –> 00:03:39,600 we also have this point

127 00:03:39,610 –> 00:03:40,089 here.

128 00:03:40,100 –> 00:03:41,179 The important one

129 00:03:41,979 –> 00:03:43,160 and I would call it

130 00:03:43,169 –> 00:03:44,960 FB minus in a

131 00:03:44,970 –> 00:03:46,070 short notation.

132 00:03:47,000 –> 00:03:48,589 Therefore, I would also include

133 00:03:48,600 –> 00:03:50,020 the point here on the right

134 00:03:50,080 –> 00:03:51,699 for calculating the length

135 00:03:51,710 –> 00:03:52,580 of the interval.

136 00:03:53,770 –> 00:03:55,550 Well, now you should see

137 00:03:55,559 –> 00:03:57,039 we have the same problem

138 00:03:57,050 –> 00:03:57,509 here.

139 00:03:57,520 –> 00:03:58,729 We have the same problem

140 00:03:58,740 –> 00:04:00,149 on the left hand side with

141 00:04:00,160 –> 00:04:01,820 the point A, you

142 00:04:01,830 –> 00:04:03,720 see A is included in

143 00:04:03,729 –> 00:04:04,610 our interval.

144 00:04:04,619 –> 00:04:05,830 And therefore, we should

145 00:04:05,839 –> 00:04:07,309 include this full

146 00:04:07,320 –> 00:04:08,979 jump in our calculation of

147 00:04:08,990 –> 00:04:10,649 the length of this interval.

148 00:04:11,199 –> 00:04:12,360 On the right hand side, we

149 00:04:12,369 –> 00:04:13,910 ignored the full jump because

150 00:04:13,919 –> 00:04:15,300 B was not in the interval,

151 00:04:15,309 –> 00:04:17,048 but now A is in the

152 00:04:17,059 –> 00:04:17,790 interval.

153 00:04:17,798 –> 00:04:19,750 And therefore we should add

154 00:04:19,760 –> 00:04:21,670 the jump to our calculation.

155 00:04:22,320 –> 00:04:23,619 You see this immediately,

156 00:04:23,630 –> 00:04:25,619 if we change this point

157 00:04:25,630 –> 00:04:27,420 maybe to the middle here,

158 00:04:28,239 –> 00:04:29,359 then we would change the

159 00:04:29,369 –> 00:04:30,450 whole calculation of the

160 00:04:30,459 –> 00:04:32,000 length, but we wouldn’t

161 00:04:32,010 –> 00:04:33,730 change the total jump.

162 00:04:34,670 –> 00:04:36,309 Hence, the only meaningful

163 00:04:36,320 –> 00:04:38,049 way to choose a point here

164 00:04:38,059 –> 00:04:39,619 would be to choose the point

165 00:04:39,630 –> 00:04:41,369 the value at the bottom here.

166 00:04:43,140 –> 00:04:45,100 And as before, I also would

167 00:04:45,109 –> 00:04:46,559 call this in short

168 00:04:46,570 –> 00:04:47,859 by fa

169 00:04:47,869 –> 00:04:49,739 minus and then we

170 00:04:49,750 –> 00:04:51,600 include the minus sign here

171 00:04:51,609 –> 00:04:52,399 as well.

172 00:04:53,109 –> 00:04:54,880 Now this thing is now our

173 00:04:54,890 –> 00:04:56,600 new notion of the length

174 00:04:56,609 –> 00:04:57,559 of an interval.

175 00:04:58,220 –> 00:04:59,380 And of course, I should give

176 00:04:59,390 –> 00:05:00,850 it a name and we call it

177 00:05:01,000 –> 00:05:02,790 mu with index

178 00:05:02,799 –> 00:05:04,109 F of this

179 00:05:04,119 –> 00:05:04,799 interval.

180 00:05:05,790 –> 00:05:07,250 And to be more precise, I

181 00:05:07,260 –> 00:05:08,929 also add the definition

182 00:05:08,940 –> 00:05:10,459 of our FB

183 00:05:10,470 –> 00:05:12,109 minus or FA

184 00:05:12,119 –> 00:05:12,859 minus.

185 00:05:14,269 –> 00:05:16,010 As you can see, this is nothing

186 00:05:16,019 –> 00:05:17,970 else than a left limit.

187 00:05:17,980 –> 00:05:19,429 So we get closer and

188 00:05:19,440 –> 00:05:20,890 closer coming from the

189 00:05:20,899 –> 00:05:21,489 left.

190 00:05:22,299 –> 00:05:23,910 Therefore, we can write this

191 00:05:23,970 –> 00:05:25,609 as an epsilon that goes to

192 00:05:25,619 –> 00:05:27,500 zero plus, which means

193 00:05:27,510 –> 00:05:29,119 it’s an epsilon greater than

194 00:05:29,130 –> 00:05:31,000 zero that goes to zero.

195 00:05:31,299 –> 00:05:33,079 And then we subtract it

196 00:05:33,089 –> 00:05:34,140 from A

197 00:05:36,769 –> 00:05:38,510 and then you see we get back

198 00:05:38,519 –> 00:05:40,309 the points we have seen in

199 00:05:40,320 –> 00:05:41,350 our drawing here.

200 00:05:42,450 –> 00:05:43,799 And at this point, you are

201 00:05:43,809 –> 00:05:45,390 allowed to ask what

202 00:05:45,399 –> 00:05:46,790 happens if I come from the

203 00:05:46,799 –> 00:05:48,619 right hand side instead of

204 00:05:48,630 –> 00:05:49,670 the left hand side.

205 00:05:50,470 –> 00:05:51,739 And then what you get is

206 00:05:51,750 –> 00:05:53,549 an alternative way to

207 00:05:53,559 –> 00:05:54,369 write this down.

208 00:05:54,380 –> 00:05:55,970 So we have FB

209 00:05:55,980 –> 00:05:56,970 plus

210 00:05:57,559 –> 00:05:58,950 minus FA

211 00:05:58,959 –> 00:06:00,850 plus where the plus

212 00:06:00,859 –> 00:06:02,579 now means the right hand

213 00:06:02,589 –> 00:06:03,250 limit.

214 00:06:05,500 –> 00:06:07,399 Now, if you go back to our

215 00:06:07,410 –> 00:06:09,359 graph, then you see,

216 00:06:09,369 –> 00:06:10,720 now we describe

217 00:06:10,730 –> 00:06:12,459 another interval, not

218 00:06:12,470 –> 00:06:14,239 this one because

219 00:06:14,250 –> 00:06:15,920 now we ignore this jump,

220 00:06:15,929 –> 00:06:17,730 but we add this jump on the

221 00:06:17,739 –> 00:06:19,480 right, this means that

222 00:06:19,489 –> 00:06:21,029 we exactly change the

223 00:06:21,040 –> 00:06:22,010 boundaries here.

224 00:06:22,019 –> 00:06:23,959 Now we measure an interval

225 00:06:23,970 –> 00:06:25,559 where A is not

226 00:06:25,570 –> 00:06:26,420 included.

227 00:06:26,730 –> 00:06:28,019 But B is

228 00:06:29,769 –> 00:06:31,630 hence, if you want to work

229 00:06:31,640 –> 00:06:33,549 with these intervals, then

230 00:06:33,559 –> 00:06:34,880 you have to consider the

231 00:06:34,890 –> 00:06:35,950 right hand limit.

232 00:06:36,799 –> 00:06:38,269 I personally want to work

233 00:06:38,279 –> 00:06:39,339 with these intervals.

234 00:06:39,350 –> 00:06:41,130 And therefore, we don’t need

235 00:06:41,140 –> 00:06:42,309 the alternative here.

236 00:06:43,959 –> 00:06:45,739 Nevertheless, it’s very important

237 00:06:45,750 –> 00:06:47,450 to note that if you look

238 00:06:47,459 –> 00:06:48,799 at the points where the function

239 00:06:48,809 –> 00:06:50,619 F is not continuous,

240 00:06:50,989 –> 00:06:52,510 then it does not matter at

241 00:06:52,519 –> 00:06:54,359 all where the value

242 00:06:54,369 –> 00:06:55,850 of the function at this point

243 00:06:55,859 –> 00:06:57,420 is it only

244 00:06:57,429 –> 00:06:58,890 matters what the limit from

245 00:06:58,899 –> 00:07:00,290 the left hand side is.

246 00:07:00,299 –> 00:07:01,450 And what the limit from the

247 00:07:01,459 –> 00:07:02,579 right hand side is

248 00:07:03,350 –> 00:07:04,970 because these two points

249 00:07:04,980 –> 00:07:06,730 describe how large the

250 00:07:06,739 –> 00:07:07,529 jump is.

251 00:07:08,540 –> 00:07:09,290 OK.

252 00:07:09,299 –> 00:07:10,679 I said before, I want to

253 00:07:10,690 –> 00:07:12,320 work with these intervals

254 00:07:12,329 –> 00:07:14,160 here because we know

255 00:07:14,170 –> 00:07:16,119 from another video that

256 00:07:16,130 –> 00:07:17,950 they form a so called semiring.

257 00:07:18,029 –> 00:07:19,809 This is what I

258 00:07:19,820 –> 00:07:21,630 explained to you in the video

259 00:07:21,640 –> 00:07:22,279 about Carathéodory’s

260 00:07:23,130 –> 00:07:24,519 extension theory.

261 00:07:25,250 –> 00:07:27,109 And now we can apply this

262 00:07:27,119 –> 00:07:28,899 theorem to conclude

263 00:07:28,910 –> 00:07:30,709 that we can extend this

264 00:07:30,720 –> 00:07:32,119 definition to a

265 00:07:32,130 –> 00:07:33,869 measure defined on the

266 00:07:33,880 –> 00:07:35,829 full Borel sigma algebra

267 00:07:35,839 –> 00:07:36,510 of R.

268 00:07:37,549 –> 00:07:38,959 This means there is

269 00:07:38,970 –> 00:07:40,839 exactly one measure

270 00:07:40,850 –> 00:07:42,209 defined on

271 00:07:42,220 –> 00:07:43,380 B(R)

272 00:07:44,570 –> 00:07:46,470 to 0 infinity

273 00:07:48,549 –> 00:07:49,929 with the property

274 00:07:50,410 –> 00:07:51,609 that we have here.

275 00:07:52,850 –> 00:07:54,190 So maybe let’s call this

276 00:07:54,200 –> 00:07:56,119 property here star and then

277 00:07:56,130 –> 00:07:58,029 I can write star here.

278 00:07:59,100 –> 00:08:00,940 Now it’s useful to call this

279 00:08:00,950 –> 00:08:02,239 uniquely given measure

280 00:08:02,250 –> 00:08:03,880 again by

281 00:08:03,890 –> 00:08:04,869 mu F.

282 00:08:05,779 –> 00:08:07,670 And if we construct a measure

283 00:08:07,679 –> 00:08:09,579 in this way, we call it a

284 00:08:09,589 –> 00:08:11,440 Lebesgue Stieltjes measure.

285 00:08:12,170 –> 00:08:13,600 And to be more concrete,

286 00:08:13,730 –> 00:08:15,570 you call it the Lebesgue

287 00:08:15,579 –> 00:08:17,399 Stieltjes measure for the

288 00:08:17,410 –> 00:08:18,549 function F

289 00:08:20,660 –> 00:08:22,279 here, you recognize how

290 00:08:22,290 –> 00:08:23,070 strong Carathéodory’s

291 00:08:23,809 –> 00:08:25,299 extension theorem really

292 00:08:25,309 –> 00:08:25,779 is.

293 00:08:26,260 –> 00:08:28,000 You only have to find the

294 00:08:28,010 –> 00:08:29,059 measure for the

295 00:08:29,070 –> 00:08:30,869 intervals and then you get

296 00:08:30,880 –> 00:08:32,719 exactly one measure

297 00:08:32,729 –> 00:08:34,440 for all Borel sets out,

298 00:08:35,460 –> 00:08:36,109 OK.

299 00:08:36,119 –> 00:08:37,558 Now, you know how this

300 00:08:37,570 –> 00:08:38,808 construction works.

301 00:08:39,030 –> 00:08:40,359 And I would suggest that

302 00:08:40,369 –> 00:08:42,320 we now look at examples.

303 00:08:43,179 –> 00:08:44,879 An example means of course,

304 00:08:44,888 –> 00:08:46,679 we choose a monotonically

305 00:08:46,689 –> 00:08:48,528 increasing function capital

306 00:08:48,568 –> 00:08:50,239 F and then we look

307 00:08:50,249 –> 00:08:51,958 what is the associated Lebesgue Stieltjes

308 00:08:52,388 –> 00:08:53,078 measure.

309 00:08:54,109 –> 00:08:54,409 OK.

310 00:08:54,419 –> 00:08:55,690 Maybe the easiest

311 00:08:55,700 –> 00:08:57,590 example is the Lebesgue

312 00:08:57,599 –> 00:08:58,599 measure itself.

313 00:08:58,799 –> 00:09:00,169 For this, we choose

314 00:09:00,179 –> 00:09:01,369 simply the

315 00:09:01,380 –> 00:09:02,190 identity.

316 00:09:02,200 –> 00:09:03,760 So the function FX

317 00:09:03,770 –> 00:09:05,690 equals to X, then

318 00:09:05,700 –> 00:09:07,049 we don’t change the

319 00:09:07,059 –> 00:09:08,840 normal measuring of

320 00:09:08,849 –> 00:09:10,260 lengths of intervals.

321 00:09:10,909 –> 00:09:12,049 So we get out

322 00:09:12,200 –> 00:09:14,109 B minus A

323 00:09:14,340 –> 00:09:15,210 as before.

324 00:09:16,260 –> 00:09:17,919 Hence, we get out our

325 00:09:17,929 –> 00:09:19,719 ordinary Lebesgue measure.

326 00:09:21,010 –> 00:09:22,440 Another example of a

327 00:09:22,450 –> 00:09:23,520 very easy

328 00:09:23,530 –> 00:09:25,229 monotonically increasing

329 00:09:25,239 –> 00:09:27,080 function would be a constant

330 00:09:27,090 –> 00:09:27,640 function.

331 00:09:28,830 –> 00:09:30,809 So let’s choose FX is

332 00:09:30,820 –> 00:09:31,969 equal to one

333 00:09:31,979 –> 00:09:32,849 everywhere.

334 00:09:33,429 –> 00:09:34,969 Obviously, in this case,

335 00:09:34,979 –> 00:09:36,590 measuring the length of

336 00:09:36,599 –> 00:09:38,289 intervals is very boring

337 00:09:39,299 –> 00:09:40,969 because we subtract right

338 00:09:40,979 –> 00:09:41,780 from left.

339 00:09:41,789 –> 00:09:43,500 So we have here everywhere

340 00:09:43,510 –> 00:09:44,400 zero.

341 00:09:45,039 –> 00:09:46,320 Hence this is not so

342 00:09:46,330 –> 00:09:48,159 surprising we get out

343 00:09:48,169 –> 00:09:49,440 our zero measure.

344 00:09:50,299 –> 00:09:50,640 OK.

345 00:09:50,650 –> 00:09:52,580 Maybe more interesting would

346 00:09:52,590 –> 00:09:54,039 be the case where we have

347 00:09:54,049 –> 00:09:55,719 two values for the function.

348 00:09:56,119 –> 00:09:57,919 So constant function, with

349 00:09:57,929 –> 00:09:59,919 the exception of one jump,

350 00:10:00,530 –> 00:10:02,150 for example, we could choose

351 00:10:02,159 –> 00:10:04,059 zero if X is

352 00:10:04,070 –> 00:10:06,010 less than zero and

353 00:10:06,020 –> 00:10:07,780 one if X is

354 00:10:07,789 –> 00:10:09,640 greater or equal than zero.

355 00:10:10,200 –> 00:10:11,869 And please remember it does

356 00:10:11,880 –> 00:10:13,659 not matter where the equality

357 00:10:13,669 –> 00:10:14,630 sign is here.

358 00:10:14,880 –> 00:10:16,159 The measure just doesn’t

359 00:10:16,169 –> 00:10:17,359 care about this.

360 00:10:17,369 –> 00:10:19,020 The measure can’t see this

361 00:10:19,030 –> 00:10:19,950 point there.

362 00:10:20,940 –> 00:10:22,530 Now, obviously, we have the

363 00:10:22,539 –> 00:10:24,359 same result as before.

364 00:10:24,369 –> 00:10:26,169 For all intervals that

365 00:10:26,179 –> 00:10:27,609 lie completely to the left

366 00:10:27,619 –> 00:10:28,280 of zero.

367 00:10:28,289 –> 00:10:29,590 Or completely to the right

368 00:10:29,599 –> 00:10:30,320 of zero.

369 00:10:30,340 –> 00:10:31,500 And we get out again, the

370 00:10:31,510 –> 00:10:32,590 measure zero.

371 00:10:33,739 –> 00:10:35,169 Hence, the interesting

372 00:10:35,179 –> 00:10:36,809 cases would be where

373 00:10:36,820 –> 00:10:38,359 zero is inside the

374 00:10:38,369 –> 00:10:39,080 interval.

375 00:10:39,770 –> 00:10:41,460 For example, let’s look at

376 00:10:41,469 –> 00:10:42,780 the interval minus

377 00:10:42,789 –> 00:10:44,200 epsilon till

378 00:10:44,210 –> 00:10:45,039 epsilon.

379 00:10:45,239 –> 00:10:46,330 Of course, where epsilon

380 00:10:46,340 –> 00:10:48,130 is a positive number.

381 00:10:49,179 –> 00:10:50,989 Now, what do we get on the

382 00:10:51,000 –> 00:10:51,979 right hand side, we have

383 00:10:51,989 –> 00:10:53,750 one on left, we have zero.

384 00:10:53,760 –> 00:10:55,219 So we have one minus

385 00:10:55,229 –> 00:10:55,760 zero.

386 00:10:55,849 –> 00:10:57,359 So we get out one.

387 00:10:58,020 –> 00:10:59,960 And please note this holds

388 00:10:59,969 –> 00:11:01,299 for all epsilon,

389 00:11:02,000 –> 00:11:03,380 we already know a measure

390 00:11:03,390 –> 00:11:05,099 that does this and this

391 00:11:05,109 –> 00:11:06,789 is the Dirac measure at

392 00:11:06,799 –> 00:11:07,369 zero.

393 00:11:08,140 –> 00:11:10,049 Now, by the uniqueness result,

394 00:11:10,059 –> 00:11:11,380 we know that the

395 00:11:11,390 –> 00:11:13,250 extension has to be this

396 00:11:13,559 –> 00:11:14,020 Dirac measure.

397 00:11:14,799 –> 00:11:16,429 So you see we can use this

398 00:11:16,440 –> 00:11:18,229 strong result very often.

399 00:11:19,299 –> 00:11:20,750 Now for the end of the video,

400 00:11:20,760 –> 00:11:22,719 let’s look at a very general

401 00:11:22,729 –> 00:11:23,479 example.

402 00:11:24,080 –> 00:11:25,130 Therefore, let’s choose a

403 00:11:25,140 –> 00:11:26,770 general monotonically

404 00:11:26,780 –> 00:11:28,479 increasing function F

405 00:11:28,900 –> 00:11:30,200 but in addition, it should

406 00:11:30,210 –> 00:11:31,809 also be continuously

407 00:11:31,820 –> 00:11:33,760 differentiable both

408 00:11:33,770 –> 00:11:34,559 things together.

409 00:11:34,570 –> 00:11:35,840 Now means that the

410 00:11:35,849 –> 00:11:37,200 derivative is

411 00:11:37,210 –> 00:11:38,650 continuous and

412 00:11:38,659 –> 00:11:40,650 also has values in the

413 00:11:40,659 –> 00:11:41,909 non-negative

414 00:11:41,919 –> 00:11:42,679 numbers.

415 00:11:44,549 –> 00:11:46,080 On the other hand, F

416 00:11:46,090 –> 00:11:47,450 itself is of course

417 00:11:47,460 –> 00:11:48,369 continuous.

418 00:11:48,380 –> 00:11:50,250 Therefore, there are no jumps.

419 00:11:50,989 –> 00:11:52,210 This means that we don’t

420 00:11:52,219 –> 00:11:53,530 have the problem with the

421 00:11:53,539 –> 00:11:54,500 left hand limit.

422 00:11:54,510 –> 00:11:55,479 And the right hand limit

423 00:11:55,489 –> 00:11:57,210 from before, for this

424 00:11:57,219 –> 00:11:58,450 continuous function, we can

425 00:11:58,460 –> 00:12:00,450 just use the values of the

426 00:12:00,460 –> 00:12:01,049 function.

427 00:12:02,239 –> 00:12:03,640 This means the length of

428 00:12:03,650 –> 00:12:05,219 the interval is nothing

429 00:12:05,229 –> 00:12:06,190 more than

430 00:12:06,200 –> 00:12:08,030 FB minus

431 00:12:08,039 –> 00:12:08,979 FA.

432 00:12:10,380 –> 00:12:12,280 Now, if you learn calculus,

433 00:12:12,289 –> 00:12:13,979 you should immediately recognize

434 00:12:13,989 –> 00:12:14,989 this one here,

435 00:12:15,500 –> 00:12:16,960 namely, here you can

436 00:12:16,969 –> 00:12:18,700 apply the fundamental

437 00:12:18,710 –> 00:12:20,309 theorem of calculus

438 00:12:20,969 –> 00:12:22,840 which tells you this is

439 00:12:22,849 –> 00:12:24,440 the integral of the

440 00:12:24,450 –> 00:12:26,340 derivative of

441 00:12:26,510 –> 00:12:28,299 F dX where

442 00:12:28,309 –> 00:12:30,260 dx just denotes the normal

443 00:12:30,340 –> 00:12:31,200 Lebesgue measure here.

444 00:12:32,520 –> 00:12:32,880 OK.

445 00:12:32,890 –> 00:12:34,380 And this is all we need

446 00:12:34,390 –> 00:12:35,979 because you would believe

447 00:12:35,989 –> 00:12:37,929 me that we can define a

448 00:12:37,940 –> 00:12:38,979 new measure.

449 00:12:39,400 –> 00:12:40,880 So for each Borel set

450 00:12:40,890 –> 00:12:42,739 A, we define the

451 00:12:42,750 –> 00:12:44,479 measure as this

452 00:12:44,489 –> 00:12:45,400 integral.

453 00:12:45,669 –> 00:12:45,690 Yeah.

454 00:12:45,729 –> 00:12:47,520 So we send the Borel

455 00:12:47,549 –> 00:12:49,140 set to this number

456 00:12:49,150 –> 00:12:49,580 here.

457 00:12:50,030 –> 00:12:51,900 This defines us a measure

458 00:12:51,909 –> 00:12:53,619 on the Borel Sigma algebra

459 00:12:53,630 –> 00:12:54,200 of R.

460 00:12:54,739 –> 00:12:56,539 And now we apply the uniqueness

461 00:12:56,549 –> 00:12:58,419 result of Carathéodory’s

462 00:12:58,429 –> 00:12:59,859 extension theorem.

463 00:13:00,320 –> 00:13:01,780 And then we get out that

464 00:13:01,789 –> 00:13:03,700 our measure mu F

465 00:13:03,710 –> 00:13:05,520 looks exactly like this

466 00:13:05,530 –> 00:13:05,859 one.

467 00:13:07,010 –> 00:13:08,419 We know it looks like this

468 00:13:08,429 –> 00:13:10,229 for all intervals

469 00:13:10,239 –> 00:13:11,659 and therefore it should also

470 00:13:11,669 –> 00:13:13,479 look like this for each Borel

471 00:13:13,669 –> 00:13:14,109 set.

472 00:13:14,969 –> 00:13:15,419 OK.

473 00:13:15,429 –> 00:13:16,880 Here you see a very general

474 00:13:16,890 –> 00:13:18,179 result for such

475 00:13:18,950 –> 00:13:19,719 Lebesgue Stieltjes measures.

476 00:13:20,289 –> 00:13:21,859 And this part here in the

477 00:13:21,869 –> 00:13:23,460 integral is then often

478 00:13:23,469 –> 00:13:24,869 called a density

479 00:13:24,880 –> 00:13:25,559 function.

480 00:13:26,570 –> 00:13:28,349 However, that’s a thing we

481 00:13:28,359 –> 00:13:29,830 could discuss in another

482 00:13:29,840 –> 00:13:31,070 video later

483 00:13:31,940 –> 00:13:32,280 here.

484 00:13:32,289 –> 00:13:33,869 I really hope you learned

485 00:13:33,880 –> 00:13:35,789 something today and that

486 00:13:35,799 –> 00:13:37,030 you can apply these

487 00:13:37,039 –> 00:13:38,390 results in other

488 00:13:38,400 –> 00:13:39,510 applications.

489 00:13:40,260 –> 00:13:42,140 If you see a monotonically

490 00:13:42,150 –> 00:13:43,479 increasing function,

491 00:13:43,489 –> 00:13:44,840 now you can

492 00:13:44,849 –> 00:13:46,289 construct the Lebesgue Stieltjes

493 00:13:46,690 –> 00:13:48,000 measure for this function.

494 00:13:48,690 –> 00:13:50,429 Well, thank you for listening

495 00:13:50,440 –> 00:13:51,750 and see you next time.

496 00:13:51,849 –> 00:13:52,590 Bye.

Q1: Which function $F: \mathbb{R} \rightarrow \mathbb{R}$ is not monotonically increasing?

A1: $F(x) = x^2$

A2: $F(x) = x$

A3: $F(x) = x^3$

A4: $F(x) = \arctan(x)$

A5: $F(x) = 0$

Q2: For a monotonically increasing function $F: \mathbb{R} \rightarrow \mathbb{R}$, we can define a measure of an interval $\mu_F( [a,b) )$. How is it defined?

A1: $$\mu_F( [a,b) ) = \lim_{\varepsilon \rightarrow 0} F(b - \varepsilon) - \lim_{\varepsilon \rightarrow 0} F(a - \varepsilon) $$

A2: $$\mu_F( [a,b) ) = \lim_{\varepsilon \rightarrow 0} F(b + \varepsilon) - \lim_{\varepsilon \rightarrow 0} F(a + \varepsilon) $$

A3: $$\mu_F( [a,b) ) = F(b) - \lim_{\varepsilon \rightarrow 0} F(a - \varepsilon) $$

A4: $$\mu_F( [a,b) ) = F(b) - F(a) $$

Q3: Consider a monotonically increasing function $F: \mathbb{R} \rightarrow \mathbb{R}$ that is also right-continuous. What is correct?

A1: $\mu_F( (0,1] ) = F(1) - F(0)$

A2: $$\mu_F( [a,b) ) = \lim_{\varepsilon \rightarrow 0} F(b + \varepsilon) - \lim_{\varepsilon \rightarrow 0} F(a + \varepsilon) $$

A3: $\mu_F( [0,1) ) = F(1) - F(0)$

A4: $\mu_F( (0,1] ) = F(1)$