A stochastic process $X=(X_n)_{n\in\mathbb N}$ on the filtered probability space $(\Omega,\mathcal F,(\mathcal F_n)_{n\in\mathbb N},\mathbb P)$ that is a martingale has the property that $$\mathbb E[X_m\mid \mathcal F_n]=X_n \quad\quad\quad\forall m\ge n$$ ($\le$ for super-martingales and $\ge$ for sub-martingales). Now I was wondering if this implies that $$\mathbb E[X_m]=\mathbb E[X_n] \quad\quad\quad\forall m\ge n$$ (as before, $\le$ for super and $\ge$ for sub) as it would make sense intuitively.
Hence my question: are the following steps correct?
Let $m\ge n$ and set the relation $\sim$ to be $=$, $\le$ or $\ge$ if $X$ is a martingale, super-martingale or sub-martingale respectively. Then \begin{array} \mathbb E[X_m\mid \mathcal F_n]\sim X_n \quad\Rightarrow\quad \mathbb E[\mathbb E[X_m\mid \mathcal F_n]]\sim\mathbb E[X_n] \quad\Rightarrow\quad \mathbb E[X_m]\sim\mathbb E[X_n] \end{array}