Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 201 / 1 / a

Put $M_n=\mathbb{E}[Z\mid {\mathcal{F}}_n]$. Each $M_n$ is ${\mathcal{F}}_n$-measurable and integrable, with $|M_n|\le \mathbb{E}[|Z|\mid {\mathcal{F}}_n]\le 1$. Since a filtration is increasing, the tower property of conditional expectation givesThus $(M_n)$ is the conditional-expectation martingale associated with $Z$.