Theorem: For an adapted stochastic process $(X_n)_{n \in \mathbb{N}} \subset L_1(\Omega, \mathfrak{A}, P)$ the following are equivalent:
i) $(X_n)_{n \in \mathbb{N}}$ is a martingale
iii) For bounded stopping times $\sigma \leq \tau$ we have $E[X_{\tau}|\mathfrak{F}_{\sigma}] = X_{\sigma}$
There is a ii) too, but it is easier to understand. Since $\sigma,\tau$ are random, statement iii) escapes my understanding.
Can you offer some insight on what iii) means and how it is useful? Thank you