 Solution

A martingale with respect to a filtration $({\mathcal{F}}_n)$ is an adapted process: $X_n$ is ${\mathcal{F}}_n$-measurable for every $n$. It must satisfy $\mathbb{E}|X_n|<\infty$ andBy the tower property this is equivalent to $\mathbb{E}[X_m\mid {\mathcal{F}}_n]=X_n$ for all $m\ge n$. Integrability and adaptation are part of the definition, not consequences of writing a formal conditional-mean identity.