How do I prove that a martingale has a constant expected value? I can´t prove that a martingale has constant expected value.
$$
\mathbf{E}[M_t]=\mathbf{E}[M_0]
$$
Thanks people.

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, $\mathbb{E}(M_n) = \mathbb{E}(M_0)$.
