Simple question. I have a stationary ergodic process $U$ on a finite alphaet and I want to prove the Kac's Lemma (see Cover and Thomas - Elements of Information Theory (second ed.) page 445).
In the proof there is $$A_{jk}:=\{ U_{-j}=u,U_{k}=u,U_{i}\neq u \, for \, -j<i<k\}$$ where U is the process, u is a fixed character, and j,i, k are integer. There yields that the probability of the union on all j and k is 1 by ergodicity, but I really cannot see why.
Probably my problem is that I do not have a clear idea about what an ergodic stochastic process is, so any kind of help in this sense is much appreciated.
Thank you!
(Sorry for my poor English!)