I apologize if this is to specific but i've already talked to two of my professors without much success and I really need to understand this subject. The following theorem is stated in Durrett page 254
Let x be a recurrent state, and let $T=\inf \{n\geq 1: X_n =x\}$ then $\mu _x (y) = \sum_{n=0} ^\infty P_x (X_n =y, T > n)$ defines a stationary measure.
He proves it and then writes a "technical note" saying that "To show we are not cheating, we should prove that $\mu _x (y)<\infty$". Why do we need that? One of my professors (who is usually right about things) talked about it being impossible for a stationary measure to give infinite mass to a point (I'm not quite sure what mass means in that context), but I really can't find any reason (in the book anyway) for that to be true.
I could really need to ask a few short questions to a person familiar with Durrett and his chapter about Markov chains.
p.s. I have the relevant pages as a pdf which I can supply if anyone would like.
Thanks in advance,
Henrik