Is $\mathsf{X}_{n+1}$ also $\mathcal{F}_n$-measurable? (martingales)

Question

Let $(\Omega, \mathcal{F}, \mathcal{F_n}, P)$ be a filtered probability space and let $(\mathsf{X}_n,\mathcal{F_n})_{n \in \mathbb{N}_0}$ be a martingale on $(\Omega, \mathcal{F}, \mathcal{F_n}, P)$.

By definition $\mathsf{X}_n$ is $\mathcal{F}_n$-measurable.

My question: Is $\mathsf{X}_{n+1}$ also $\mathcal{F}_n$-measurable?

Answer 1

It cannot be unless $X_n$ is independent of $n$. $X_n=E(X_{n+1}|\mathcal F_n)$. If $ X_{n+1}$ is already measurable w.r.t. $\mathcal F_n$ this would give $X_{n+1}=X_n$.