Invariant initial law makes a Markov chain stationary

For every $m\ge 0$ one has $\pi K^m=\pi$ as probability measures: for $m=0$ this is $\pi K^0=\pi I=\pi$, and if $\pi K^m=\pi$ then associativity of iterated kernel composition gives $\pi K^{m+1}=(\pi K^m)K=\pi K=\pi$ by [F1].

For fixed $m$ the family ${\mathcal{D}}_m:=\{B\in {\mathcal{E}}^{\otimes {\mathbb{N}}_0}:{\mathbb{P}}_X({\theta }^{-m}B)={\mathbb{P}}_X(B)\}$ is a lambda-system: it contains $E^{{\mathbb{N}}_0}$, and ${\theta }^{-m}(B^c)=({\theta }^{-m}B{)}^c$ together with ${\theta }^{-m}\left({\bigcup }_l{B}_l\right)={\bigcup }_l{\theta }^{-m}{B}_l$ for pairwise disjoint families give closure under complements and disjoint countable unions by additivity of the probability measure ${\mathbb{P}}_X$ of [F6].

Fix $m\ge 0$, times $0\le n_0<\cdots <n_r$ and bounded measurable $f_0,\dots ,f_r$. [F3] applied to the shifted tuple $(m+n_0,\dots ,m+n_r)$, whose successive gaps are again $n_1-n_0,\dots ,n_r-n_{r-1}$, expresses $\mathbb{E}{\prod }_{j=0}^r{f}_j(X_{m+n_j})$ as ${\int }_E\pi (dx){\int }_E{K}^{m+n_0}(x,d{x}_0)f_0(x_0){\prod }_{j=1}^r{\int }_E{K}^{n_j-n_{j-1}}(x_{j-1},d{x}_j)f_j(x_j)$, while [F3] applied to $(n_0,\dots ,n_r)$ gives the same expression with $K^{n_0}$ in place of $K^{m+n_0}$; associativity [F4] gives $K^{m+n_0}=K^m{K}^{n_0}$, so step 1.1 gives $\pi K^{m+n_0}=(\pi K^m)K^{n_0}=\pi K^{n_0}$, the two outermost integrals over $\pi$ coincide, and all remaining factors are identical.

Taking $f_j={\mathbf{1}}_{A_j}$ in step 2.1 shows $\mathbb{P}(X_{m+n_j}\in A_j,0\le j\le r)=\mathbb{P}(X_{n_j}\in A_j,0\le j\le r)$ for all measurable $A_0,\dots ,A_r$; measurable rectangles form a pi-system generating ${\mathcal{E}}^{\otimes (r+1)}$, and by [F5] two probability measures agreeing on it agree on the whole product sigma-algebra, so $\mathcal{L}(X_{m+n_0},\dots ,X_{m+n_r})=\mathcal{L}(X_{n_0},\dots ,X_{n_r})$, which is the asserted shift invariance of every finite-dimensional law.

Let ${\mathbb{P}}_X:=\mathcal{L}((X_n{)}_{n\ge 0})$ be the canonical path law on $(E^{{\mathbb{N}}_0},{\mathcal{E}}^{\otimes {\mathbb{N}}_0})$ [F6], and fix $m\ge 0$; for the cylinder $C=\{z:z_{n_j}\in A_j,0\le j\le r\}$ one has ${\theta }^{-m}C=\{z:(z_{m+n_j}{)}_j\in {\prod }_j{A}_j\}$, so step 3.1 gives ${\mathbb{P}}_X({\theta }^{-m}C)=\mathbb{P}(X_{m+n_j}\in A_j\text{ for all }j)=\mathbb{P}(X_{n_j}\in A_j\text{ for all }j)={\mathbb{P}}_X(C)$.

By step 4.1 the family ${\mathcal{D}}_m$ contains every finite-dimensional cylinder, and cylinders form a pi-system generating ${\mathcal{E}}^{\otimes {\mathbb{N}}_0}$, so [F5] gives ${\mathcal{D}}_m={\mathcal{E}}^{\otimes {\mathbb{N}}_0}$; hence ${\mathbb{P}}_X({\theta }^{-m}B)={\mathbb{P}}_X(B)$ for every measurable $B$, and for $m=1$ the left shift preserves the canonical path law.

Boundary and axiom cases: if $E$ is a singleton the canonical path law is the point mass at the constant path and every shift preserves it; if $r=1$ and $m=0$ the identities in steps 2.1–3.1 are trivial; the equivalence between the finite-dimensional and canonical formulations is proved in both directions, steps 1.1–3.1 giving the finite-dimensional statement and steps 4.1–5.1 the path-space statement; and AC [A1] enters exactly through the finite-dimensional-law supplier [F3], whose statement itself assumes Choice, while the induction, the rectangle comparison and the lambda-system computation are ordinary measure-theoretic algebra.