= Solution
Relative to a <probability space> $(\Omega,\mathcal F,\mathbb P)$ and a <filtration> $(\mathcal F_n)$, the real <stochastic process> $(M_n)$ is a <martingale> if it is <adapted>, each $M_n$ is an <integrable random variable>, and
$$
\boxed{\mathbb E[M_{n+1}\mid\mathcal F_n]=M_n\quad\text{almost surely for every }n\geq0.}
$$
Here <adapted> means $M_n$ is $\mathcal F_n$-measurable, and integrability means $\mathbb E|M_n|<\infty$. By the <tower property of conditional expectation>, the equivalent multi-step condition is $\mathbb E[M_j\mid\mathcal F_i]=M_i$ for all $i\leq j$. The <filtration> is part of the definition; specifying only the marginal means is insufficient.
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