$(X_n), n \in \mathbb{N}$ is a stochastic process.
I saw in one definition of Martingale that $E [X_{n+1} |X_0 , X_1 , . . . , X_n ] = X_n \quad a.s., \forall n \geq 0.$
- I understand what "almost surely" itself means. But I was wondering how to interpret the above usage of "almost surely" in the definition of Martingale? Are $X_0 , X_1 , . . . , X_n$ seen as random variables or their given deterministic values?
- Plus, I don't see the definition of Martingale on its Wikipedia article relies on "almost surely". Is it really required?
Thanks and regards!