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Lecture 13

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1. Continuity of monotone functions.

Definition 1. A function on an open0ÐBÑinterval is if wheneverÐ+ß ,Ñ § ‘ increasing B Ÿ B 0ÐBÑ Ÿ 0ÐB Ñw w then .

It is if whenever thendecreasing B Ÿ Bw

0ÐBÑ   0ÐB Ñw .It is if it is either everywheremonotone

increasing or everywhere decreasing

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Theorem 1. A monotone function on Ð+ß ,Ñhas at most a countable number ofdiscontinuities.

Proof (sketch): Assume the contrary. Thenthere would be an uncountable number ofpoints such that .B 0ÐB Ñ 0ÐB Ñ !

But then there would be a sub-intervalÒ-ß .Ó § Ð+ß ,Ñ such that there is anuncountable number of discontinuous pointsin .Ò-ß .Ó

[using a simple argument by contradiction]

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But then the increase would be an0Ð.Ñ 0Ð-Ñuncountable sum of positive numbers, whichis easily shown to be . This gives a∞contradiction.

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2. Differentiability of montone functions

Definition 2: A collection of closedYbounded nontrivial intervals covers a set Iin the sense of Vitali if for every , thereB − Iis an arbitrarily short interval M − Ycontaining .B

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Vitali Covering Lemma: Let be a set ofIfinite outer measure in , and be a‘ Ycollection of closed bounded intervalscovering in the sense of Vitali. Then forIany there is a finite disjoint% !subcollection such thatÖM × §3 3œ"

8 Y. %‡

33ÐI µ ∪ M Ñ .

[That is, you can cover by a finitealmost Inumber of disjoint intervals in collection ]Y

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[Note: here we use the proof of Fitzpatrick,though this proof is at points hard to follow]

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Proof: Since has finite outer measure,Ithere is an open set of finiteb ¨ Imeasure.

We will show that we can assume without lossthat the sets in are all contained in Y b À

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Step 1: Claim: to prove the CoveringLemma we may assume without loss that allthe sets in are contained in the open setYb.

Proof: Note that for all there is anB − Iinterval with and (sinceM − B − M M §Y bthese intervals are allowed to be arbitrarilyMsmall).

Now for each , keep only intervals B − I M − Ythat are in . Replace by this newb Ysmaller collection.

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Then we still have a Vitali covering of , andIclearly it suffices to prove the CoveringLemma for this smaller collection.

Thus we assume that all sets in are withinYthe open set .b

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Step 2: We select an infinte sequence of setsM ß M ßá §" # Y as follows:

First we choose any .M −" Y

If the Covering Lemma is proved andI § M"we are done. If not, then letY Y 9" "œ ÖM − À M ∩ M œ ×Þ

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[Note is non-empty since there is anY"

B − I µ M which is a positive distance awayfrom (since is closed), so withM M b M − YB − M M M ∩ M œ and is small enough that .]" 9

Let . Let be such that= œ jÐMÑ M −" # "M−supY"

Y

jÐM Ñ   = Î# M M# " # ". Note is disjoint from .

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Again if we are done.M ∪ M ¨ I" # Otherwise,let . LetY Y 9# 3œ ÖM − À M ∩ M œ ß 5 œ "ß #×= œ jÐMÑ M −# $ #

M−supY#

. Let be such thatY

jÐM Ñ   = Î#Þ$ #

Again if we are done.M ∪ M ∪ M ¨ I" # $

Otherwise let Y Y 9$ 5œ ÖM − À M ∩ M œ ß5 œ "ß #ß $× = œ jÐMÑ M −, and . Let $ % $

M−supY$

Y

be such that jÐM Ñ   = Î#Þ% $

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We continue this way to generate an infinitecollection Note since theyM ß M ßá − Þ" # Y

are disjoint and in , we have ,b5œ"

∞

5j M ∞

so that by above .5œ"

∞

5= ∞

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Step 3: Claim: every finite sub-union 5œ"

8

5M

has the property that the remainder-5œ8"

∞

55 blown up 5 times covers ofM all

I µ ∪ M5œ"

8

5.

Proof: let .B − I µ ∪ M5œ"

8

5

Let , with chosen so thatB − M − MY

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.M ∩ ∪ M œ5œ"

8

5 9

This is possible since is closed and∪ M5œ"

8

5

B  ∪ M b M −5œ"

8

5, so a sufficiently short Y

such that .M ∩ ∪ M œ5œ"

8

5 9

Then must intersect some (with ),M M R 8R

for otherwise we would have forjÐMÑ Ÿ =Rall which is impossible since .R = Ä !5

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So let be the first number such thatR 8M ∩ M Á M ∩ M œR R"9 9. Then , sojÐMÑ Ÿ = jÐM Ñ   = Î#R" R R". But .

Since and intersect, and , itM M jÐMÑ Ÿ #jÐM ÑR R

must be that . Thus .M § &M B − &MR R

Since for each , we haveB − I µ ∪ M5œ"

8

5

B − &M R 8R for some , we conclude

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B − ∪ &M5œ8"

∞

5, as desired. This proves the

claim.

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Step 4: Claim: The Covering Lemma nowholds.

Proof:Now as above choose so that8

5œ8"

∞

5jÐM Ñ Î&% .

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We know that , soI § ∪ M ∪ ∪ &MŒ Œ5œ" 5œ8"

8 ∞

5 5

I µ ∪ M § ∪ &M5œ" 5œ8"

8 ∞

5 5, and

. . %‡ ‡

5œ"

8

5 5

5œ8"

∞ŒI µ ∪ M Ÿ & ÐM Ñ ,

as desired.

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3. Derivatives

Definition 3: If is a function and is an0ÐBÑ Binterior point of its domain, the upperderivative of at is0 B

H0ÐBÑ œ ß0ÐB >Ñ 0ÐBÑ

>lim sup2Ä! l>lŸ2

” •

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while the islower derivative

H0ÐBÑ œ ß0ÐB >Ñ 0ÐBÑ

>lim inf2Ä! l>lŸ2

” •We say is at if0 BdifferentiableH0ÐBÑ œ 0ÐBÑÞH

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4. Lebesgue theorem on differentiability

[Note: here we use the proofs of Fitzpatrick]

Lemma 2: Let be increasing on .0ÐBÑ Ò+ß ,ÓThen for each ,α !

7 B − Ð+ß ,Ñ H0ÐBÑ   Ÿ Ò0Ð,Ñ 0Ð+ÑÓß"‡˜ ™¸ αα

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and

7 ÖB − Ð+ß ,Ñ lH0ÐBÑ œ ∞× œ !Þ‡

Proof: Step 1: DefineI œ ÖB À H0ÐBÑ   × ! Þα α α α. Choose w

Let be the collection of intervals inY Ò-ß .ÓÐ+ß ,Ñ 0Ð.Ñ 0Ð-Ñ   Ð. -ÑÞ such that αw

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Step 2: Claim: Since on , is a H0   Iα Yα

Vitali covering of .Iα Indeed, every B − Iα

must be in an interval of the formÒ-ß .Óabove.

Step 3: By Vitali covering there is a disjointsubcollection such thatÒ- ß . Ó −5 5 Y

7 I µ ∪ Ò- ß . Ó ‡

5œ"

8

5 5Π%.

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Step 4: Claim: The Theorem now holds:Proof:

We now have

I § ∪ Ò- ß . Ó ∪ I µ ∪ Ò- ß . Ó Þα Œ Œ5œ" 5œ"

8 8

5 5 5 5

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So

7 ÐI Ñ Ÿ Ð. - Ñ ‡

5œ"

8

5 5α %

Ÿ 0Ð. Ñ 0Ð- Ñ "

α%

w5œ"

8

5 5c d

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Ÿ 0Ð,Ñ 0Ð+Ñ ß"

α%

wc d

since is increasing.0

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Since and are abitrary we can let% α α ! w

% α αÄ ! Ä and , and the result still holds.w

Proof of second part is easy.

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Lebesgue's theorem: If is monotone on0ÐBÑthe open interval , then it isÐ+ß ,Ñdifferentiable a.e. on .Ð+ß ,Ñ

Proof:

Step 1: We can assume is increasing, and0wolog assume is bounded.Ð+ß ,Ñ

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Step 2: The set of points where is not0differentiable can be written as

ÖB À H0ÐBÑ 0ÐBÑ×H

œ ÖB À H0ÐBÑ 0ÐBÑα " H for some

rationals α " ×

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œ ∪ ÖB À H0ÐBÑ 0ÐBÑ×α "

α "ß rational

ðóóóóóóóóóóóóóóóñóóóóóóóóóóóóóóóòα " H

Iα "ß

´ ∪ I . (1)α "

α "

ß

α "ß rational

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Step 3: Note that to show that is0differentiable a.e., it suffices to show that7 ÖB À H0ÐBÑ 0ÐBÑ× œ !‡ H .

Since the above union in (1) is a countableunion, it suffices to show that for7 I œ !‡

ßα "

aribitrary α "ß Þ

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Step 4: Let By definition of outerI œ I Þα "ß

measure, let be an open set (union ofbopen intervals) such that andI § b7 Ð Ñ 7 ÐIÑ ‡ ‡b %, which can be done forany choice of .% !

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Step 5: Let be the set of intervalsYÒ-ß .Ó § 0Ð.Ñ 0Ð-Ñ Ÿ Ð. -Ñb " such that . We claim these form a Vitali covering of .I

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Proof: Since on , for every point H0 I B"in there are arbitrarily small intervalsIÒ-ß .Ó ® B for which the approximatederivative is arb. close to0Ð.Ñ0Ð-Ñ

.- ¸ 0ÐBÑH

H0 ", so that

0Ð.Ñ 0Ð-Ñ

. - Þ"

In fact since this is true for arbitrary ,B − Ithe intervals form a Vitali covering ofÒ-ß .ÓI.

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Step 6: By the Vitali covering lemma thereare a finite number of disjoint intervalse fÒ- ß . Ó §5 5 5œ"

8 Y such that

7 I µ ∪ Ò- ß . Ó ‡

5œ"

8

5 5Π%. (2)

Thus

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c d5œ"

8

5 50Ð. Ñ 0Ð- Ñ

Ð. - Ñ Ÿ 7Ð Ñ Ÿ Ð7 ÐIÑ Ñ" " b " %5œ"

8

5 5‡ (3)

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Step 7: Applying the previous Lemma to on0the interval , we have (sinceÒ- ß . Ó5 5

H 0ÐBÑ IÑ2 α on

7 ÐI ∩ Ò- ß . ÓÑ Ÿ 0Ð. Ñ 0Ð- Ñ Þ"‡

5 5 5 5α

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Step 8: Therefore by above(2)

[note that is measurable so below]I 7 œ 7‡

7 ÐIÑ Ÿ 7 ÐI ∩ Ò- ß . ÓÑ ‡ ‡

5œ"

8

5 5 %

Ÿ 0Ð. Ñ 0Ð- Ñ Þ"

α%c d

5œ"

8

5 5

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Step 9: Therefore, by (3)

7 ÐIÑ Ÿ 7 ÐIÑ ‡ ‡"

α% %

œ 7 ÐIÑ Þ" "

α α% %‡

[note there is a small typo in the textbook: "α

is mistakenly replaced by there]"α

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Since this holds for all , it follows that% !

7 ÐIÑ Ÿ 7 ÐIÑ Ê 7 ÐIÑ œ !‡ ‡ ‡"α , as desired.

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5. Fundamental theorem of calculus

Corollary 1: Let be increasing on .0 Ò+ß ,ÓThen (now defined a.e.) is integrable, and0 w

(+

,w0 .B Ÿ 0Ð,Ñ 0Ð+ÑÞ

[Note this proof is a bit different fromFitzpatrick's proof]

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Proof:

Step 1: Extend to by defining0ÐBÑ B ,0ÐBÑ œ 0Ð,Ñ in this interval.

We know that a.e., the derivative exists,0 ÐBÑw

i.e.

lim2Ä!

w0ÐB 2Ñ 0ÐBÑ

2œ 0 ÐBÑ

exists.

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Letting , define2 œ "Î8

1 ÐBÑ ´ 0ÐBÑ ´0ÐB "Î8Ñ 0ÐBÑ

"Î88 "Î8Diff

œ 8 0ÐB "Î8Ñ 0ÐBÑ Þc dNote since is increasing.1 ÐBÑ   ! 08

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Step 2: Then by Fatou:

( ( (ðóñóò+ + +

, , ,

8 81ÐBÑ œ 1 ÐBÑ Ÿ 1 ÐBÑlimlimit exists a.e. by Lebesgue Theorem

lim

œ 8 Ò0ÐB "Î8Ñ 0ÐBÑÓ.Blim (+

,

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œ 8 0ÐB "Î8Ñ.B 8 0ÐBÑ.Blim ( (+ +

, ,

œ 8 0ÐBÑ.B 8 0ÐBÑ.Blim ( (+"Î8 +

,"Î8 ,

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œ 8 0ÐBÑ .Blim – —( (, +

,"Î8 +"Î8

œ 8 † 0Ð,Ñ 8 0ÐBÑ.B"

8lim – —(

+

+"Î8

[since by construction on ]0 œ 0Ð,Ñ Ò,ß , "Î8Ó

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œ 0Ð,Ñ 8 0ÐBÑ.Blim (+

+"Î8

Ÿ 0Ð,Ñ 0Ð+Ñ

[since is increasing]0 Þ

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6. Functions of bounded variation

[Note this material is covered as inFitzpatrick's writeup]

For any function on , define a0ÐBÑ Ò+ß ,Ópartition .+ œ B B á B œ ,! " 5

Let T œ ÖB ß B ßá ß B ×Þ! " 5

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Define the variation

XZ Ð0 ß T Ñ œ l0ÐB Ñ 0ÐB Ñl3œ"

5

3 3"

Definition 4: Define the of astotal variation 0

XZ Ð0Ñ œ ÖZ Ð0 ß T Ñ À T Ò+ß ,Ó×Þsup a partition of

Def. 5: If we say that has X ∞ 0 boundedvariation.

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Definition 6: The function for total variation 0is , where is theX ÐBÑ œ XZ Ð0 Ñ 00 Ò+ßBÓ Ò+ßBÓ

function restriced to the interval 0 Ò+ß BÓÞ

Note that it is easy to show that is anX ÐBÑ0

increasing function.

Also easy to show that if then+ Ÿ ? Ÿ ,

XZ Ð0 Ñ œ XZ Ð0 Ñ XZ Ð0 ÑÞÒ+ß,Ó Ò+ß?Ó Ò?ß,Ó (4)

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Also it follows that if , then+ Ÿ ? @ Ÿ ,l0Ð@Ñ 0Ð?Ñl Ò?ß @Ó is the variation on underthe trivial partition, so:

0Ð?Ñ 0Ð@Ñ Ÿ l0Ð@Ñ 0Ð?Ñl Ÿ XZ Ð0 Ñ œÒ?ß@Ó by (4)

XZ Ð0 Ñ XZ Ð0 ÑÒ+ß@Ó Ò+ß?Ó

Thus , so0Ð?Ñ XZ Ð0 Ñ Ÿ 0Ð@Ñ XZ Ð0 ÑÒ+ß?Ó Ò+ß@Ó

that is a monotone increasing0ÐBÑ X ÐBÑ0

function of B

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Jordan's Theorem: A function is of0bounded variation on if and only if it isÒ+ß ,Óthe difference of two monotone increasingreal-valued functions.

Proof: ( ) If has bounded variation thenÊ 0write

0ÐBÑ œ 0ÐBÑ X ÐBÑ X ÐBÑàc d0 0

thus is a difference of two increasing0ÐBÑfunctions.

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Conversely, if is the0ÐBÑ œ 1ÐBÑ 2ÐBÑdifference of two increasing functions, wehave

Z Ð0 ß T Ñ œ l0ÐB Ñ 0ÐB Ñl3œ"

8

3 3"

Ÿ l1ÐB Ñ 1ÐB Ñl l2ÐB Ñ 2ÐB Ñl3œ"

3

3 3" 3 3"

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Ÿ Z Ð1ß T Ñ Z Ð2ß T ÑÞ

Thus:

XZ Ð0Ñ Ÿ XZ Ð1Ñ XZ Ð2Ñ ∞.

Since for the monotone function on ,1 Ò+ß ,ÓXZ Ð1Ñ œ l1Ð,Ñ 1Ð+Ñl since every partitiongives this variation and similarly for .2Hence is finite and has boundedXZ Ð0Ñ 0variation.

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Definition 7: The above decomposition of abv function into0ÐBÑ œ 1ÐBÑ 2ÐBÑmonotone increasing functions is called theJordan decomposition of .0ÐBÑ

Corollary 2: If has bounded variation on0Ò+ß ,Ó 0 ÐBÑ then exists for almost allw

B − Ò+ß ,Ó 0 ÐBÑ Ò+ß ,ÓÞ, and is integrable on w