Measure, Integration & Real Analysis · Problem 12.14

Probability Measures — Sheldon Axler

Mathematics — Measure Theory · distribution-function-continuity-equivalences

Problem

Suppose

(\Omega, \mathcal{F}, P)

is a probability space and

X

is a random variable. Prove that the following are equivalent. •

\widetilde{X}

is a continuous function on

\mathbf{R}

. •

\widetilde{X}

is a uniformly continuous function on

\mathbf{R}

. •

P(X = t) = 0

for every

t \in \mathbf{R}

. •

(\widetilde{X} \circ X)\widetilde{\phantom{X}}(s) = s

for all

s \in [0, 1]