I'm trying to learn about Martingales with specific focus on combinatorial problems. However i'm far from an expert in algebra and am having some trouble understanding the basic idea. I will write the exact paragraph that i'm reading and then some areas where i'm struggling.
-- Given a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and an increasing sequence of sub-$\sigma$ fields $\mathcal{F}_{0}= \{\emptyset,\Omega\} \subseteq \mathcal{F}_{1} \subseteq ... \subseteq \mathcal{F}_{n}=\mathcal{F}$, a sequence of random variables $X_{0},X_{1},...,X_{n}$ (with finite expectations) is called a martingale if for each $k=0,...,n-1$, $\mathbb{E}(X_{k+1}|\mathcal{F}_{k})=X_{k}$. In this case (with a finite sequence), every martingale is obtained from a random variable $X$ by taking $X_{k}=\mathbb{E}(X|\mathcal{F}_{k})$, $k=0,...,n$. Then $X_{0}=\mathbb{E}(X)$ and $X_{n}=X$. Also, we always have $\mathbb{E}(X_{k+1})=\mathbb{E}(X_{k})$. --
Ok so i'm trying to keep everything within a combinatorial context specifically to random graphs.
So i know a sigma field on some set is a collection of subsets satisfying certain set theoretic properties (i.e for every subset containment of it's compliment). Can anyone give a combinatorial example of a sequence of increasing sigma fields?
Secondly i'm unsure how $X_{k}$ is defined. What exactly does $\mathbb{E}(X|\mathcal{F}_{k})$ mean? Is it the restriction of the random variable $X$ to the sigma field $\mathcal{F}_{k}$?
I think if someone could perhaps provide a concrete simple combinatorial example it would help me greatly. I appreciate any help!
