Let $(M_t)_t$ a stochastic process. Let $(\mathcal F_t)_t$ an adapted filtration. The process $(M_t)_t$ is called a martingales if
1) $\mathbb E|M_t|<\infty $
2) $\mathbb E[M_t\mid \mathcal F_s]=M_s$ where $s\leq t$.
My problem is the 2) in the definition. Indeed, since the filtration is adapted, then $M_t$ is $\mathcal F_t-$measurable for all $t$. By definition of a filtration, $\mathcal F_s\subset \mathcal F_t$ provided $s\leq t$. Now, I know that if $X$ is $\mathcal G-$measurable, then,$$\mathbb E[X\mid \mathcal G]=X\mathbb E[\boldsymbol 1\mid \mathcal G]=X.$$
Now, I'm trying to do a link of this and the fact that $\mathbb E[M_t\mid \mathcal F_s]=M_s$, but I don't see. So, obviously $M_t$ is not $\mathcal F_s-$measurable, otherwise, we would have that $\mathbb E[M_t\mid \mathcal F_s]=M_t$, but is there an idea near that behind the martingale ?