Q: Martingales and different definitions Are there any differences between the following definitions of Martingales and if so what are they? 


*

*Let $(X_{i})_{i=1}^{n}$ and $(Z_{i})_{i=1}^{n}$ be sequences of random variables then $(X_{i})$ is a martingale if $\mathbb{E}(X_{i}|Z_{1}...Z_{i})=X_{i-1}$ for all $i$. 

*Let $\mathcal {F}_{0} \subseteq \mathcal{F}_{1} \subseteq...\subseteq \mathcal{F}_{n}$ be a sequence of $\sigma$-subfields. Then $(X_{i})$ is a martingale with respect to $\mathcal{F}_{i}$ if $\mathbb{E}(X_{i}|\mathcal{F}_{i})=X_{i-1}$ for all $i$
In addition i was wondering what $\mathbb{E}(X_{i}|\mathcal{F}_{i})$ actually means, as it combines two notions of probability that are usually distinct from my previous experience in the sense that a random variable is a function defined on a probability space. I appreciate any help thanks!

A: (1) is the definition of the martingale in the special case where the filtration $\mathscr{F}_i$ is the $\sigma$-algebra generated by the random variables $Z_1,\ldots,Z_i$. (2) is the "general" case.
