Consistency between two definitions of conditional expectation/martingale: $$E(X_{n+1}|\mathcal{F}_n)=X_n\implies\forall A\in\mathcal{F}_n, E(X_{n+1}|A)=X_n?$$ An interpretation from Durrett: Given $\mathcal{F}_n$ means you know what event in it has occurred. But this doesn't seem to hold because the left hand side is always a number while the right hand side is a random variable.
However the below is true since $\sigma(X_n)\subset\mathcal{F}_n$ $$E(X_{n+1}|\mathcal{F}_n)=X_n\implies E(X_{n+1}|X_n)=X_n?$$
Consider $$E(X_{n+1}|\mathcal{F}_n)=X_n\implies\forall k, E(X_{n+1}|X_n=k)=k?$$
By definition from Durrett: \begin{align} E(X_{n+1}|X_n=k)&=\frac{\int_{\{X_n=k\}}X_{n+1}dP}{P(X_n=k)}\\ &=\frac{\int E(X_{n+1}|\mathcal{F}_n)\mathbb{1}_{\{X_n=k\}}dP}{P(X_n=k)}\\ &=\frac{\int E(X_{n+1}\mathbb{1}_{\{X_n=k\}}|\mathcal{F}_n)dP}{P(X_n=k)}\\ &=\frac{E(X_{n+1}\mathbb{1}_{\{X_n=k\}})}{P(X_n=k)}\\ &=?? \end{align}