I know it was asked before, but I still don't understand the answers in the post and I think I am missing some key arguments to understand but not sure which. I am reading Cover and Thomas "Elements of information theory", and on page 445 the author prooves Kac Lemma. First we got a
Let $\{ U_{i} \}_{i \in \mathbb{Z}}$ be a stationary ergodic process on a countable alphabet. for any $u$ s.t $p(u)>0$
next, in the beginning of the proof:
define the event:
for $j=1, 2, 3, ...$ and $k = 0, 1, 2, ... $ $A_{jk} := \{U_{-j}=u, U_{l} \neq u \ \forall -j<l<k, U_{k}=u\}$
in the following sentence the author writes
by ergodicity, the probability $Pr(\cup_{j,k}A_{jk})=1$
now, for the last day I tried to prove this claim using stationary egrodic process properties. I must say that this is for a course in information theory and not ergodic theory (which I never learned), but we had a couple of lessons about ergodicity especially in the light of stationary stochastic processes.
My attemp so far was to try a rewrite $A_{jk} =: [uLu]_{-j}^k$ so if $T: X^{\mathbb{Z}}\rightarrow X^{\mathbb{Z}}$ is the trasformation defined as $\pi_i(T(x))=x_{i+1}$ ($\pi_i$ is the projection of the i-th' coordinate, so T is the left shift transformation), since the process is stationary ergodic, T is ergodic and $T([uLu]_{-j}^k)=[uLu]_{-j-1}^{k-1}$ and $T^{-1}([uLu]_{-j}^k)=[uLu]_{-j+1}^{k+1}$.
So after trying playing with the definition of ergodicity on those sets in order to prove $Pr(\cup_{jk}A_{jk})=1$ I understand I am missing something clearly. Is it part of my definitions of stochastic stationary ergodic process? Or it's actually a hard-try to prove with simple tools? or I'm just missing something simple?!
Thanks.