MHW
Lv 5

# Measure theoretic question?

Let X be a random variable and define g on the real numbers by

g(t) = E[ e^(itX) ] for t in R

(ie. g is the characteristic function of X).

(i) Suppose |g(t)| = 1 for some non zero t. Prove that there exists a real constant s such that

P(X = s + 2kπ/t for some integer k) = 1.

(ii) Suppose there exist non zero t and irrational u such that

|g(t)| = |g(ut)| = 1.

Use the result you proved in (i) to show that X is constant with probability 1.