I have the following problem:
Let $p$ and $q$ be two probability measures on a countable set $E$ with $p\neq q$ and $q(x)>0$ for all $x\in E.$ Further let $(X_n)$ be a sequence of i.i.d. random variables on $E$ with distribution $q$.
Proof that $Y_n = \prod_{k=1}^{n} p(X_k)/q(X_k)$ is a positive martingale that converges to $0$ a.s.
I started with proofing the martingale property $\mathbb{E}[Y_n|\mathcal{F_{n-1}}] = \mathbb{E}[Y_{n-1} p(X_n)/q(X_n)|\mathcal{F_{n-1}}] = Y_{n-1}\mathbb{E}[p(X_n)/q(X_n)|\mathcal{F_{n-1}}]$,
but I'm not sure how to get $\mathbb{E}[p(X_n)/q(X_n)|\mathcal{F_{n-1}}] = 1$.
Any help with that and in regards to the convergence is appreciated.