Measure Theory Notes

8/11/2019 Measure Theory Notes

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1 Measure Theory

Definition 1.1. (Measure). Let be a set andA a -algebra of subsets of. The pair (, A) is call a measurable space. A set function onAis called ameasure if is an extended real-valued function satisfying

(1) (A) 0 for all A A(2) () = 0(3) IfA1, A2,... are disjoint sets inA, then

n

An

=n

(An).

The triple (, A, ) is called a measure space.Theorem 1.1. Properties of Measure). Let (, A, ) be a measure spaceand letA, B beA-measurable sets. Then:(a) If(A)< and A B, then (B A) = (B) (A)(b) Monontonicity: A B (A) (B).(c) Countable Subadditivity: if{An}n=1 A, then

n

An

n

(An)

(d) If{An}n=1 Awith A1 A2... and (Ak)< for somek , then

n=1

An

= lim

n(An).

(e) If{An}n=1 Awith A1 A2..., then

n=1

An

= lim

n(An).

Corollary 1.2. More Properties of Measure). Let (, A, ) be a measurespace and

{An

}

n=1

A. Then

(i) (lim infn An) lim infn (An)(ii) (lim supn An) lim supn (An), as long as (

n=kAn)


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Theorem 1.3. (Linear Combinations of Measures). Let (, A) be a mea-surable space,

{n

}n=1a sequaence of measures on

A, and

{n

}n=1a sequence

of nonnegative real numbers. Then if we define

n=1 nn onAbyn=1

nn

(A) =

n=1

nn(A),

then we have defined a new measure onA.Definition 1.2. (Complete Measure Space). A measure space (, A, ) issaid to be completeif, for everyA-measurable setA with (A) = 0,

B A B A.Every measure space can be extended to a complete measure space.

Theorem 1.4. (Completion of a Measure Space). If (, A, ) is a measurespace,

2 Measurable Functions

2.1 Real and Complex valued Functions

Definition 2.1. (Measurable Function). Let (, A) be a measurable space,the function f : Ris said to be anA-measurable function if

f1(O) A, for every open setO R.A complex valued function g :

Cis said to be an

A-measurable function if

both its real part, [g], and its imaginary part, [g], are A-measurable functions.Theorem 2.1. (Measurability). Let (, A) be a measurable space and f : Ra real-valued function. The following are equivalent:(a) f isA-measurable.(b) For eacha R, f1((, a)) A(c) For eacha R, f1((, a]) A(d) For eacha R, f1((a, )) A(e) For eacha R, f1([a, )) ATheorem 2.2. (Measurable Functions Form an Algebra). Let (, A) bea measurable space. The collection of real-valuedA-measurable functions on forms an algebra. That is, if f, g : R areA-measurable functions, and R, then(1) f+ g isA-measurable

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(2) f isA-measurable

(3) f g isA-measurableThis also holds for complex valued functions.

Theorem 2.3. (More Properties of Measurable Functions). Let (, A)be a measurable space and f : RanA-measurable function. Then:

(i) Ifg : R Ris a Borel measurable function, then g f isA-measurable.(ii) f+, f and|f| :=f+ + f areA-measurable.

2.2 Extended Real-valued Functions

see p.179 in Mcdonald.

Theorem 2.4. (More Properties). Let (, A) be a measurable space. Letf, g : R be extended real-valuedA-measurable functions, and{fn}n=1 asequence of extended real-valuedA-measurable functions, fi: R. Then:

(i) f g andf g areA-measurable.(ii)|f| :=f f isA-measurable.

(iii) supn fn and infn fn areA-measurable.(iv) lim supn and lim infn areA-measurable.(v) If{fn}n=1 converges pointwise, then limn fn isA-measurable.

(vi) If (,A

, ) is a complete measure space,{

fn}

n=1 a sequence of extended

real-valued or complex real-valuedA-measurable functions, and fn f-almost everywhere, then f isA-measurable.

Theorem 2.5. (Egorovs Theorem). Let (, A, ) be a measure space, andlet E Abe a set with (E) 0,there is a measurable set A Esuch that

(E A)< and fn funiformly onA

2.3 Approximation with Simple Functions

Recall that the indicator function (or characteristic function) IEof a set E isgiven by

IE=

1 ifx E0 ifx =E

Note that IEis measurable if and only ifE is measurable.

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Definition 2.2. (Simple Function). Asimple functionfis a function whichassumes a finite number of finite values. When fis simple, it has the represen-

tation

f(x) =Nk=1

akIEk(x),

wherea1,...,aNare the distinct values taken by fon the disjoint sets E1,...,EN.Note thatfis measurable iff each Ek is measurable.

Theorem 2.6. (Approximation by Simple Functions).

(i) Any (measurable) function f 0 can be represented as the limit of anincreasing sequence of (measurable) simple functions. One such represen-tation is given by:

fk(x) = j2k if j2k

f(x)< j+12k

, j = 0,...,k2k

1

k iff(x) kwherefk(x)f(x) as k . Note that we can also write each simplefunctionfk(x) as

fk(x) =Kj=0

j

2kIf[ j

2k, j+12k

)(x) + kIfk(x)

whereK:= k2k 1(ii) Any (measurable) functionfcan be represented as the limit of a sequence

of (measurable) simple functions. In particular, apply the previous resultto approximate f+ and f respectively by increasing simple functions,

and then:f+k (x) fk (x) f+(x) f(x) = f(x)

3 Convergence of Functions

For all that follows, let (, A, ) be a measure space. Let {fn}n=1be a sequenceof (complex-valued) measurable functions on E A.Theorem 3.1. (The Set of Convergence). If{fn} is a sequence of functions,the set on which convergence occurs is characterized by

{x : limn

fn(x) = f(x)} =

m=1

n=1

k=n

x: |f(x) fk(x)| < 1

m=

m=1

liminfn

x: |f(x) fn(x)| < 1

m

= limm

liminfn

x: |f(x) fn(x)| < 1

m

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Definition 3.1. (Uniform Convergence).{fn} converges uniformlytofon if

limn

supx

{|fn(x) f(x)|}

= 0.

That is, fn funiformly on if for all >0,N such thatsupx

{|fn(x) f(x)|} , n N.

4 Abstract Lebesgue Integral

4.1 Integral of Nonnegative Functions

4.1.1 Convegence of Integrals of Nonnegative Functions

Theorem 4.1. (Monotone Convergence Theorem). Suppose 0 fn fae onE. Then for each E A, Efnd Ef d.Corollary 4.2. Let f,g, {fn}n=1 0 be extended real-valuedA-measurablefunctions and letE A. Then

(i)E(f+ g)d=

E

f d +E

gd

(ii)E

n=1 fnd=

n=1

E

fnd

(iii) If{E}n=1 A are pairwise disjoint and E :=

n=1 En, thenE

f d =n=1

En

f d.

(iv) If{E}n=1 A is an increasing sequence of sets, En E:=n=1 En, and

iff

0 is an extended real-valued

A-measurable function, then Ef d=

limnEn fd.

4.2 Integral of General Functions

Definition 4.1. (Integral of Extended Real-Valued Function). Let (, A, )be a measure space. Letfbe an extended real-valuedA-measurable functionon E A. Then the Lebesgue integral off overEwith respect to is definedby

E

f(x)d(x) =

E

f d:=

E

f+d E

fd,

as long as at least one of the integrals is finite (in which case the Lebesgueintegral is said to exist). If both integrals are finite,

Ef+d,

Efd


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Theorem 4.3. (Properties of Lebesgue Integrals). Let (, A, ) be ameasure space. Let f, g be extended real-valued

A-measurable functions on

E A.(i) f L1(E, ) |f| L1(E, ).

(ii)| Ef d| E |f|d(iii) If|f| |g| -ae in E, and if g L1(E, ), then f L1(E, ) and

E|f|d

E|g|d.

(iv) Iff L1(E, ), thenf is finite-ae inE.(v) Iff = g -ae in E and

E

f d exists, thenE

gd exists andE

f d =E

gd.

(vi) IfEf d exits and R, then Efd exists and Efd= Ef d.

(vii) Iff, g L1(E, ), then f+g L1(E, ) and E(f+g)d) = Ef d+Egd.

(viii) Iff 0 and m g Mae on E, then

m

E

fd E

fgd ME

f d

Theorem 4.4. (Tchebyshevs Inequality). For >0,

({x E: |f(x)| }) 1p

{f>}

fp

Proposition 4.1. (Contructing New Measures). Let (,A

, ) be a mea-sure space. Ifg : R+ is a nonnegativeA-measurable function, andE A,then the set function defined by

(E) :=

E

gd

is a measure onA. Moreover, iff isA-measurable, then

f d=

fgd.

Example 4.1. If denotes Lebesgue measure on (R, B(R)), then the standardnormal distribution can be written as

(E) =E

12 e

x2/2

(dx), E B(R),and for measurable fwe can compute

E

f(x)(dx) =

E

f(x) 1

2ex

2/2(dx)

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4.3 Complex Integrals

Whenf is complex-valued on , we have

|f| =

([f])2 + ([f])2.Proposition 4.2. Letfbe a complex-valued function on . Then:

(i)|f| | [f]| + |[f]|(ii)|[f]| |f| and|[f]| |f|

(iii) Iff isA-measurable, then so is|f|.(iv) LetE A. Then|f| L1(E) [f], [f] L1(E), in which case

E

f d= E

[f]d + iE

[f]d.

5 Convegence of Integrals

For all of the following theorems, let (, A, ) be a measure space. Let{fn}n=1be a sequence of (complex-valued) measurable functions on E A.Theorem 5.1. (Monotone Convergence Theorem).

(i) Suppose fnfae on E. IfL1(E, ) such that fn ae on E forall n, then

E

fnd E

f d.

(ii) Suppose fnfae on E. IfL1(E, ) such that fn ae on E forall n, then Efnd Ef d.

Theorem 5.2. (Fatous Lemma).

(i) If L1(E, ) such that fn ae on Efor all n, thenE

(lim infn

fn)d liminfn

E

fnd

Thus iffn fae on E,E

f d liminfn

E

fnd

(ii) If L1(E, ) such that fn ae on Efor all n, then

E

(limsupn fn)d limsupnE

fnd

Thus iffn fae on E,E

f d limsupn

E

fnd

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Theorem 5.3. (Dominated Convergence Theorem). If L1(E, )such that

|fn

| ae on Efor all n, then

(i) E

(lim infn

fn)d liminfn

E

fnd

limsupn

E

fnd E

(lim supn

fn)d

(ii) Thus iffn fae on E, then f L1(E, ) andE

fnd E

fd and limn

E

|fn f|d= 0.

There is a more general result that allows for a sequence of bounding func-tions.

Theorem 5.4. (Extended Dominated Convergence Theorem). Suppose

{n}, L1(E, ) such that|fn| n ae on Efor all n, such that n ae ,and

E

|n|d E

||d. Iffn fae on E, then f L1(E, ) andE

fnd E

f d and limn

E

|fn f|d= 0

Corollary 5.5. (Bounded Convergence Theorem). Suppose (E)


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Corollary 5.8. (Continuity). Suppose thatf L1(, ).

(i) Suppose {E}n=1 Ais an increasing sequence of sets, so thatEn E:=

n=1 En. Then E

fd= limn

En

fd.

(ii) Suppose{E}n=1 Ais a decreasing sequence of sets, so that En E:=n=1 En. Then

E

fd= limn

En

fd.

Corollary 5.9. Suppose that f L1(, ), and{E}n=1 A is a sequence ofpairwise disjoint sets, and E:=

n=1 En. Then

E

f d=

n=1

En

fd.

5.2 Some Approximation Corollaries

For all of the following theorems, let (, A, ) be a measure space.Corollary 5.10. Suppose that f L1(, ).

(i) fis finite ae in .

(ii) Continuity: 0, >0 such that (E)< E |f|d < .(iii) 0, there exists a bounded,A-measurable function g such that

|f g|d <

(iv) 0, there exists a simple,A-measurable function h such that

|f h|d <

Proof: see Athreya pg. 66.

5.3 Convegence in Measure

For all of the following theorems, let (,

A, ) be a measure space, and let

{fn}n=1 be a sequence of (complex-valued) measurable functions on E A.Definition 5.1. (Convergence in Measure). The sequence{fn} is said toconverge in measure to the measurable function f onE if, > 0,

limn

({x E: |f(x) fn(x)| }) = 0.

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This is denoted by fn f on E. In other words, fn f on E iff > 0,

N such that

({x E: |f(x) fn(x)| })< , n N.Definition 5.2. (Cauchy in Measure). The sequence{fn} is said to beCauchy in measure if, >0,

limn,m

({x E: |fm(x) fn(x)| }) = 0.

Proposition 5.1. (Convergent Implies Cauchy). If the sequence{fn} isconvergent in measure, then it is Cauchy in measure. On complete measurespaces, this is an iff.

Theorem 5.11. (Convergence in Measure Results). Suppose f,g, {fn}, {gn}are measurable onE A.

(i) Fix a R. Iffn f, ({x: fn(x)> a}) ({x: f(x)> a})(ii) Iffn

f andgn g, then (fn+ gn) (f+ g)(iii) If in addition(E)< , then fngn f g.(iv) If(E)< ,fn f,gn pw g, andg= 0 ae onE, thenfn/gn f/g.(v) Suppose fn

f and let g : C C be a continuous function. Theng fn g f . For example:

fn f, (fn)p fp

Note that fngn fg = (fn f)(gn g) + f(gn g) + g(fn f), a usefulidentity for proving results about products.

Theorem 5.12. (Connection with Convergence Almost Everywhere).Statements of convergence are understood to be onE A.

(i) First note thatfnae f iff > 0,

limn

k=n

{x E: |f(x) fk(x)| }

= 0.

(ii) Iffnae f, thenfn f(as long as (E)< ).

(iii) If fn f, then there exists a subsequence {fnj}j=1 on E satisfying

fnjae f asj . Even more specifically, there is a subsequence{fnk}

satisfying

{x E: |fnk(x) fnk+1(x)| 2k} 2k

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Theorem 5.13. (Fatous Lemma for Convergence in Measure). Supposethat

{fn

}n=1

0 is a sequence of (complex-valued)

A-measurable functions such

thatfn fon . Then for each E A,E

f d limn

E

fnd.

Theorem 5.14. (DCT for Convergence in Measure). Suppose that {fn}n=10 is a sequence of (complex-valued)A-measurable functions such thatfn fon . Suppose further that for someg L1(, ), g 0, we have|fn| g aefor every n. Then

E

f d= limn

E

fnd.

Thus, the Dominated Convergence Theorem still holds even when almosteverywhere convergence is replaced by convergence in measure. (See Mcdonaldpg. 205 for proof)

6 Lp Spaces and Convergence

6.1 Definitions

Let (, A, ) be a measure space. For 0< p < andE A, the classLp(E, )denotes the collection of measurable (complex-valued) functions f :E C forwhich

E |f|p < . In particular

Lp(E, ) :=

f :

E

|f|p <

, 0< p < .

We write

fp,E :=

E

|f|p 1

p

,

or in the typical case whereE ,

fp :=

|f|p 1

p

.

For simplicity, we often omit reference to the region of integration, since theresults stated will typically hold whether the integral is taken over the entiredomain , or over some subsetE A. In this case, we will still use the simplernotation

fp := |f|p1p

when the region of integration is either understood or is not a primary concern.To define L spaces, we introduce the following:

ess supEf:= inf{ R :({x E: f(x)> }) = 0},

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if a finite exists, otherwise if({x E : f(x) > }) > 0 for all R, letess supEf :=

. Thus,

ess supEf= inf{M [, ] :f(x) Mae inE}.We now write

f= f,E:= ess supE|f|.WhenF 0.

(i) Then

iixi

i

i ii(xi)

i

i

(ii) In particular, ifk

i=1 i = 1, then

i

ixi

i

i(xi).

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Proof: see Wheeden and Zygmund pg. 119.

Theorem 6.3. (Jensens Integral Inequality). Letf, g (, A, ) be finiteae on a set E A. Suppose that f g and g are integrable on E, g 0 andE

gd >0. Suppose : R Ris convex over an interval containing the rangef(E) off.

(i) Then

EfgdEgd

E(f)gd

Egd

(ii) In particular, ifE

gd = 1,

E

fgd

E

(f)gd.

Corollary 6.4. (Useful Inequalities). Fix an integer k 1.(i) Let a1,...,ak Rand 1,...,k 0 such that

ki=1 i = 1. Then

ki=1

iexp{ai} exp

ki=1

iai

(ii) Letb1,...,bk 0 and 1,...,k 0 such thatk

i=1 i= 1. Then

ki=1

ibiki=1

bii ,

and in particular

1

k

ki=1

bi

ki=1

bi

1k

(iii) For anya, b R, and 1 p < ,|a + b|p 2p1(|a|p + |b|p)

Proof: Follow from Jensens Inequality.

Theorem 6.5. (Holders Inequality). Let 1 < p


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Corollary 6.6. (Holders Inequality: p= 1, q=). Letf L1(, ),andg

L(, ). Then since

|f g

| |f

|g

ae, we have

f g1 |fg|d f1g.

Corollary 6.7. (Discrete Holders Inequality). Letk N. Leta1,...,akR, b1,...,bk R and c1,...,ck 0. If 1 < p 0 we have

E

|f fn|p 0 as n andE

|fn|p M, thenE

|f|p M.

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Proposition 6.3. Letf , {fn} Lp(), for some p >0.

(i) fn Lp

f f fnp 0.(ii) Iff fnp 0, thenfnp fp (convergence in Lp implies conver-

gence of the norms).

(iii) Suppose 1 p < . Iffnp fp andfn fae, thenf fnp 0(sofn

Lp f).

Proposition 6.4. Fix any 1p 0.The sequence{fn} is said to converge weakly tof if

fngd

fgd for all

g Lp.Proposition 6.5. Letf , {fn} Lp(), for some 1< p < . Iffnp M forall n, then

fngd

fgd for all g Lq, with 1/p + 1/q= 1.

Proposition 6.6. Letf , {fn} Lp(), for some 1 p . Iffn Lp

f, thenfn fweakly to f.

6.6 Hilbert Spaces

Proposition 6.7. (Parallelogram law for L2).

f+ g2 + f g2 = 2f2 + 2g2

Theorem 6.10. (Cauchy-Schwarz Inequality). Letf , g L2(, ). Then |f g|d

|f|2d

12

|g|2d 1

2

,

that is

f g

1

f

2

g

2.

Proof: Follows from Holders inequality with p = q= 2.

Proposition 6.8. Letf , {fn} L2(). Iffn converges weakly to f inL2, thatis if

fngd

fgdfor all g L2, and iffn2 f2, thenfn L

2 f. (Soweak convergence plus convergence of the norms implies convergence in L2).

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7 Product Spaces and Iterated Integrals

Theorem 7.1. (Minkowskis Integral Inequality). Let 1 p


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(i) IfTis injective and continuous, then T is openT1 is continuous fromT(X) toX.

(ii)

Theorem 9.1. (The Open Mapping Theorem) IfX, Yare Banach spaces,andT L(X, Y), then T(X) is a closed subspace ofY T is open.

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