How do I prove that a martingale has a constant expected value?
It holds for any sigma-algebra $\mathcal{F}$ that
$$\mathbb{E}[ \mathbb{E}(X \mid \mathcal{F}) ] = \mathbb{E}(X).$$
Note that this does not require that $\mathcal{F}$ and $X$ are independent. Since a martingale satisfies
$$\mathbb{E}(M_{n+1} \mid \mathcal{F}_n) = M_n,$$
we get
$$\mathbb{E}(M_{n+1}) = \mathbb{E}[\mathbb{E}(M_{n+1} \mid \mathcal{F}_n)] = \mathbb{E}(M_n)$$ for all $n \in \mathbb{N}$. Hence, $\mathbb{E}(M_n) = \mathbb{E}(M_0)$.