Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 27J / d / i / Solution

Set $Z=e^{iuX}$. Then $|Z|=1$ almost surely andThus $Z=\mathbb{E}Z$ almost surely. Its constant value has modulus one, so it equals $e^{i\theta }$ for some $\theta \in \mathbb{R}$. Consequently the unit modulus of a characteristic function criterion gives