• Johns Hopkins University

• EN.625 728

Theory of Probability (625.728)

Spring 2026

Page: 1 of 4

Due: 3/9/26

Theory of Probability (625.728) { Spring 2026

All random variables are assumed to be defined on (Ω; A; µ). We prove Chapter 3,

Theorem 5(i):

Theorem 5 (i): Assume that the sequence of random variables fXng converges mutually

in measure. Then there is a subsequence fXnkg of fXng and a random variable X0 such that

fXnkg converges to X0 a.e. and in measure.

Proof): Choose an integer n1 ≥ 1 such that

µ jXm - Xn1j ≥ 12 ≤ 1 2 for m ≥ n1 :

Choose n2 > n1 such that

µ jXm - Xn2j ≥ 212 ≤ 212 for m ≥ n2 :

Using induction, we can choose a sequence 1 ≤ n1 < n2 < n3 < · · · such that

µ jXm - Xnkj ≥ 21k  ≤ 21k for m ≥ nk :

In particular, nk+1 > nk, so that

µ jXnk+1 - Xnkj ≥ 21k  ≤ 21k for k ≥ 1:

Consider the sets

Ak = ! : jXnk+1 - Xnkj ≥ 21k  k ≥ 1 ;

and define

A =

1 \ k

=1

1 [ ν=k

A

ν :

The set A has measure zero. Indeed, using countable subadditivity, we have for each k ≥ 1,

Theory of Probability (625.728)

Spring 2026

Page: 2 of 4

Due: 3/9/26

0 ≤ µ(A) ≤ µ

1 [ ν=k

A

ν! ≤ 1

2k-1 ; (1)

which implies that µ(A) = 0. This follows from the continuity property of measures.

Observe that for each ! = 2 A the following infinite series converges