Problem set 5, submit solutions by 17.10.2012

Prof. N. Kajino, Probability Theory WS 2012/2013

Problem set 5, submit solutions by 17.10.2012

TheProblems below will be discussed in the tutorial on 19.10.2012.

(TheExercises are additional and will be discussed only if time permits.) Throughout this problem set,.X;M; /denotes a given measure space.

Problem 1.26([7, Chapter 1, Exercise 9]). Let˛2.0;1/, letf WX !Œ0;1beM- measurable and supposeR

Xf d2.0;1/. Find the limit (with log1 WD 1^˛WD 1)

nlim!1

Z

X

nlog 1C.f =n/^˛ d:

(Note that log.1Cx/ 0for anyx 2 Œ0;1. If˛ 1, then log.1Cx^˛/˛x for anyx2Œ0;1. If˛ < 1, then Fatou’s lemma (Theorem 1.27) applies.)

Exercise 1.27(30 points). Letf W X ! Œ 1;1. Prove that the following three conditions are equivalent:

(1)f isM-measurable.

(2) There existM-measurable functionsf1; f2WX !Œ 1;1such thatf1 f f2onX andf1Df2-a.e.

(3) There exists aM-measurable functionf0WX !Œ 1;1such thatf0Df -a.e.

((1) ) (2): Iff isM-simple, then (2) can be proved by using the definition of M. For a generalM-measurablef, take non-decreasing sequences¹s_n^˙º¹nD1ofM- simple functions converging tof^˙, use (2) fors_n^˙and take lim sup_n!1, lim infn!1.) Problem 1.28. Letp2.0;1/and letf 2L^p./. Prove that

nlim!1

Z

X

ˇˇf f1_¹jfjnº

ˇˇ^pdD0: (1.81)

Problem 1.29. Letp; q2.0;1/,p < q, and letf WX !Œ0;1beM-measurable.

Prove that

Z

X

f^pd 1=p

Z

X

f^qd 1=q

.X /^{.q p/=pq}: (1.82) By Problem 1.29, if.X / <1, thenL^q.X; /L^p.X; /for anyp; q2.0;1/ withp < q.

Problem 1.30(Minkowski’s inequality). Letp2Œ1;1/and letf; gWX !Œ0;1be M-measurable. Prove that

Z

X

.f Cg/^pd 1=p

Z

X

f^pd 1=p

C Z

X

g^pd 1=p

: (1.83)

(Assumingp > 1, apply H¨older’s inequality tof .f Cg/^{p 1}andg.f Cg/^{p 1}.) For the next problem, we need the following definition.

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Definition. Letf WX !RandfnWX !R,n2N, beM-measurable. We say that

¹fnº¹nD1converges in-measure tof if and only if for any"2.0;1/,

nlim!1 ¹x2X j jfn.x/ f .x/j "º

D0: (1.84)

Problem 1.31. Letf WX !RandfnWX!R,n2N, beM-measurable.

(1) Let p 2 .0;1/and suppose limn!1kfn fkL^p./ D 0. Prove that¹fnº¹nD1

converges in-measure tof. (Use Problem 1.20-(2) with'.x/Dx^p,x2Œ0;1.) (2) Suppose that¹fnº¹nD1converges in-measure tof. Prove that there exists a strictly increasing sequence¹nkº¹kD1Nsuch that limk!1fnk.x/Df .x/for-a.e.x2X.

(Choosen_k 2Nso that ¹x 2X j jfnk.x/ f .x/j 2 ^kº

2 ^kand use Problem 1.11-(2) to show that Xnlim inf_k!1¹x2Xj jfnk.x/ f .x/j< 2 ^kº

D0.) Problem 1.32. LetA2 M, and define a measurej^AonMjA D ¹B\Aj B 2Mº byj^A WDjMj^A (note thatMjA M). Letf W X !Œ 1;1beM-measurable.

Prove thatR

Xf1Adexists if and only ifR

Afj^Ad.j^A/exists, and in this case Z

A

f dWD Z

X

f1AdD Z

A

fj^Ad.j^A/: (1.85) (Modify the proof of Theorem 1.43. It suffices to prove (1.85) forf^˙and hence when f 0. Show first for non-negativeM-simple functions by using Proposition 1.25 and then use Proposition 1.19 and Theorem 1.24 for general non-negativef.)

According to Problem 1.32,R

Af dcould alternatively be defined as the integral offj^Awith respect toj^ADjMj^A, therestriction oftoA.

Exercise 1.33. LetN be a-algebra in X such that N M, and let f W X ! Œ 1;1beN-measurable. Prove thatR

Xf dexists if and only ifR

Xf d.jN/exists (note thatjNis a measure on.X;N/), and in this case

Z

X

f dD Z

X

f d.jN/: (1.86)

Exercise 1.34. Letf W X !Œ0;1beM-measurable and-integrable. Prove that, for any"2.0;1/there existsı 2.0;1/such thatR

Af d < "for anyA2Mwith .A/ < ı. (Proof by contradiction. Problem 1.11-(2) can be used.)

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