Application $\pi$-$\lambda$ lemma one-sided Markov shift

Question

Let $(S_k^{\mathbb{N}},\Sigma_k^{\mathbb{N}},m,\tau)$ be the probability preserving transformation of the one-sided Markov shift, where $\Sigma_k^{\mathbb{N}}$ is the $\sigma$-algebra generated by the cylinder sets.

Let $$ B_\infty:=\bigcap\limits_{n=1}^{\infty}\tau^{-n}\Sigma_k^{\mathbb{N}} $$ I have to prove that $$ \mathcal{L}=\{B: m(A\cap B)=m(A)m(B)\quad\forall A\in B_{\infty} \} $$ is a $\lambda$-system, i.e.

• $L\in\mathcal{L}$ implies $L^c\in\mathcal{L}$
• $(L_n)\subset\mathcal{L}$ implies $\bigcup\limits_n L_n\in\mathcal{L}$

Answer 1

For a fixed $A\in B_\infty$, consider $\mathcal L_A:=\{B: m(A\cap B)=m(A)m(B)\}$. It suffices to show that each $\mathcal L_A$ is a $\lambda$ system, that is,

• $L\in\mathcal L_A$ implies that $L^c\in\mathcal L_A$ and
• if $L_n\in \mathcal L_A$ for each $n$, and $L_n\subset L_{n+1}$ (which is missing in the OP), then $\bigcup_{n\geqslant 1}L_n\in \mathcal L_A$.

For the first item, note that $m(A\cap L^c)=m(A)-m(A\cap B)$, then use the assumption that $L\in\mathcal L_A$.

For the second one, compute the following limits $$\lim_{n\to\infty}m(A\cap L_n)\mbox{ and } \lim_{n\to\infty}m(L_n).$$