Conditions for a process to be a martingale

Hi !

I'll give the introduction but it's not the main problem.

We have E a countable space of states. Q is an irreducible transition matrix. Let $\displaystyle (\Omega,\mathcal{F},(X_n)_n,\mathbb{P}_x)$ ($\displaystyle \mathbb{P}_x$ denotes the initial probability) be the canonical Markov chain with the transition matrix Q.

Let $\displaystyle f ~:~ E\to [0,\infty)$ such that $\displaystyle Qf=f$

Prove that $\displaystyle (f(X_n))_n$ is a martingale.

So I have no problem showing the equality of the conditional expectation, and showing that it's measurable wrt the filtration.

But I don't know how to show that $\displaystyle f(X_n)$ is integrable...

So my questions :

Just to be sure : does it mean to show that $\displaystyle \mathbb{E}_x(f(X_n))$ is finite ?

A friend said that there were martingales with an infinite expectation... How so ?

Is something positive always integrable ?

Would the hypothesis that $\displaystyle (X_n)_n$ is a Markov chain be useful ? Or the hypothesis that Q is irreducible ?

Erm... That's all for now.

Thanks :)