Skip to document
Welcome to StudocuSign in to access the best study resources
Guest user
0followers
HomeMy LibraryAsk AI
  • You don't have any recent items yet.
My Library
  • You don't have any modules yet.
  • You don't have any books yet.
  • You don't have any Studylists yet.

Probability and Measure 2017-2018 Example Sheet 2

Module

Probability and Measure

8 Documents
Students shared 8 documents in this course
Academic year: 2017/2018
Uploaded by:
Anonymous Student
This document has been uploaded by a student, just like you, who decided to remain anonymous.
Newcastle University

Comments

Please sign in or register to post comments.

Students also viewed

Related documents

Preview text

E. Breuillard Michaelmas 2017

Probability and Measure 2

3.1 (X,A) be a measurable space. Suppose that a simple functionfhas two representations

f=

∑m

k=

ak 1 Ak=

∑n

j=

bj 1 Bj.

Show that, for any measureμ, ∑m

k=

akμ(Ak) =

∑n

j=

bjμ(Bj).

hint: forε= (ε 1 ,... , εm)∈ { 0 , 1 }m, defineAε=Aε 11 ∩.. .∩AεmmwhereA 0 k=AckandA 1 k=Ak. Define similarlyBδforδ∈{ 0 , 1 }n. Then setfε,δ=

∑m k=1εkakifAε∩Bδ 6 =∅andfε,δ= 0 otherwise. Show then that ∑m

k=

akμ(Ak) =

∑

ε,δ

fε,δμ(Aε∩Bδ)

3.2μandνbe finite Borel measures onR. Letf be a continuous bounded function onR. Show thatfis integrable with respect toμandν. Show further that, ifμ(f) =ν(f) for all such f, thenμ=ν.

3.3 be a real-valued integrable function on a measure space (X,A, μ). LetFbe a family of subsets fromA, which is stable under intersection, containsXand generates theσ-algebraA. Suppose thatμ(f 1 F) = 0 for all subsetsF∈F. Show thatf= 0μ-a.

1

2

3.4 a non-negative integer-valued random variable. Show that

E(X) =

∑∞

n=

P(X≥n).

Deduce that, ifE(X) =∞andX 1 , X 2 ,.. .is a sequence of independent random variables with the same distribution asX, then, almost surely, lim supn(Xn/n) = ∞. (hint: use the second Borel-Cantelli lemma) Now suppose thatY 1 , Y 2 ,.. .is any sequence of independent identically distributed real-valued random variables withE|Y 1 |=∞. Show that, almost surely, lim supn(|Yn|/n) =∞, and moreover lim supn(|Y 1 +···+Yn|/n) =∞.

3.5 that the product of the Borelσ-algebras ofRd 1 andRd 2 is the Borelσ-algebra ofRd 1 +d 2. Give an example to show that this is no longer the case if the word Borel is replaced by Lebesgue.

3.6 that the function sinx/xis not Lebesgue integrable over [1,∞) but that integral

∫N

1 (sinx/x)dx converges asN→∞. Show that the functionf(x) :=x 2 sin(x 12 ) is continuous and differentiable at every point of [0,1] but its derivative is not Lebesgue integrable on this interval.

3.7 that, asn→∞, ∫∞

0

sin(ex)/(1 +nx 2 )dx→0 and

∫ 1

0

(ncosx)/(1 +n 2 x

32 )dx→ 0.

3.8 differentiable functions onRwith continuous derivativesu′andv′. Suppose thatuv′andu′vare integrable onRandu(x)v(x)→0 as|x|→∞. Show that ∫

R

u(x)v′(x)dx=−

∫

R

u′(x)v(x)dx.

4

3.13μandνbe probability measures on (E,E) and letf:E→[0, R] be a measurable function. Suppose thatν(A) =μ(f 1 A) for allA∈E. Let (Xn:n∈N) be a sequence of independent random variables inEwith lawμand let (Un:n∈N) be a sequence of independent uniformly distributed on [0, R] random variables. Set

T= min{n∈N:Un≤f(Xn)}, Y =XT.

Show thatY has lawν. (This justifies simulation by rejection sampling.)

Was this document helpful?

Probability and Measure 2017-2018 Example Sheet 2

Module: Probability and Measure

8 Documents
Students shared 8 documents in this course

University: University of Cambridge

Was this document helpful?
E. Breuillard Michaelmas 2017
Probability and Measure 2
3.1. Let (X, A) be a measurable space. Suppose that a simple function fhas two representations
f=
m
X
k=1
ak1Ak=
n
X
j=1
bj1Bj.
Show that, for any measure µ,
m
X
k=1
akµ(Ak) =
n
X
j=1
bjµ(Bj).
hint: for ε= (ε1, . . . , εm)∈ {0,1}m, define Aε=Aε1
1∩. . . ∩Aεm
mwhere A0
k=Ac
kand A1
k=Ak.
Define similarly Bδfor δ∈ {0,1}n. Then set fε,δ =Pm
k=1 εkakif Aε∩Bδ6=∅and fε,δ = 0 otherwise.
Show then that m
X
k=1
akµ(Ak) = X
ε,δ
fε,δµ(Aε∩Bδ)
3.2. Let µand νbe finite Borel measures on R. Let fbe a continuous bounded function on R.
Show that fis integrable with respect to µand ν. Show further that, if µ(f) = ν(f) for all such
f, then µ=ν.
3.3. Let fbe a real-valued integrable function on a measure space (X, A, µ). Let Fbe a family
of subsets from A, which is stable under intersection, contains Xand generates the σ-algebra A.
Suppose that µ(f1F) = 0 for all subsets F∈ F. Show that f= 0 µ-a.e.
1

Students also viewed

Related documents