Homework IV
STAT 559: Measure theory
Hua Zheng
April. 28, 2015
1
Exercise 2.3.2
1.1
a
Proof. To show this, we simply need an example satisfying that Xn X but Xn a.e. X.
Suppose that = [0, 1), A all Borel sets in and is Lebesgue measure on [0, 1). Then den

Homework II
STAT 559: Measure theory
Hua Zheng
April. 13, 2015
1
Exercise 1.2.1
Proof. To prove Exercise 1.2.1, we need to prove the following claims:
1. (1) cfw_A : A1 A A2 with A1 , A2 A and (A2 \ A1 ) = 0
(2) = cfw_A N : A A and N (some B) A having (B)

Homework I
STAT 559: Measure theory
Hua Zheng
April. 8, 2015
1
Exercise 1.1.1
Proof. We rst prove [C1 ] [C2 ]. Since
C1 [C2 ] =
cfw_F : F is a f lied of subsets of f or which C2 F ,
which implies that any -eld in [C2 contains C1 . Thus the intersection of