 Solution

A martingale relative to a filtration $({\mathcal{F}}_n)$ is an adapted sequence of integrable random variables with $E[X_{n+1}\mid {\mathcal{F}}_n]=X_n$ almost surely for every $n$. Adapted means $X_n$ is ${\mathcal{F}}_n$-measurable. Iterating gives $E[X_m\mid {\mathcal{F}}_n]=X_n$ for all $m\ge n$.