the version of Doobs Optional sampling theorem that I Have is as follows:
Let $\sigma, \tau$ be bounded stopping times with $\sigma \leq \tau.$ Then we have
(i) If $(X_{t})$ is a martingale, then \begin{align*} \mathbb{E}[X_{\tau} \mid F_{\sigma}] = X_{\sigma} \quad \text{and thus} \quad \mathbb{E}[X_{\tau}] = X_{0} \end{align*} (ii)If $(X_{t})$ is a sub-martingale (super-martingale), then \begin{align*} \mathbb{E}[X_{\tau} \mid F_{\sigma}] \geq X_{\sigma} \quad (\mathbb{E}[X_{\tau} \mid F_{\sigma}] \leq X_{\sigma}) \end{align*}
My question is now: Why is it not if and only if? Because every konstant is a stopping time and therefore I could easily get the conditions for a martingale.