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 ( denotes the initial probability) be the canonical Markov chain with the transition matrix Q.

Let such that

Prove that 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 is integrable...

So my questions :

Just to be sure : does it mean to show that is finite ?

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

Is something positive always integrable ?

Would the hypothesis that is a Markov chain be useful ? Or the hypothesis that Q is irreducible ?

Erm... That's all for now.

Thanks :)