A geometrically ergodic chain has an all-start standard-Borel observation factor

Define $\lambda =\frac{1}{2}\pi +\frac{1}{2}{\sum }_{k\ge 0}{2}^{-k-1}\mu P^k\mathrm{.}$ This is a probability by [F8], $\pi ,\mu \ll \lambda$, and $\lambda P\ll \lambda$: if $\lambda (A)=0$, then $\pi (A)=0$ and $\mu P^k(A)=0$ for every $k$, which also makes $\lambda P(A)=0$. Thus $\lambda P^j\ll \lambda$ for every $j$. Apply [F2] to code the pair $(f,M)$ into one real observable, and apply [F1] to it. The resulting countably generated $\mathcal{G}\subseteq \mathcal{F}$ is kernel-stable and contains both $f$ and $M$, since the coding has a Borel inverse.

Enumerate generators $A_j$ of $\mathcal{G}$ and set $c(x)={\sum }_{j\ge 1}2\cdot 3^{-j}{1}_{A_j}(x)$, taking values in the closed Cantor subset $C$ of $[0,1]$. Its coordinate digits recover every $A_j$, so $\sigma (c)=\mathcal{G}$. The target $C$ is standard Borel by [F9]; no Borel-image assertion about $c(X)$ is needed. By [F3], write $f={\phi }_0\circ c$ and $M=m_0\circ c$ for Borel extended-real functions on $C$. Put ${\lambda }_c=c_{\ast }\lambda$, ${\pi }_c=c_{\ast }\pi$ and ${\mu }_c=c_{\ast }\mu$.

Push the probability $\lambda (dx)P(x,dy)$ to $(c(x),c(y))\in C^2$ and disintegrate by [F4], obtaining a Borel kernel $T:C\to C$. For each $B$ in a fixed countable generating algebra of $C$, kernel stability and [F3] factor $P(x,c^{-1}B)$ through $c(x)$. Its value is the conditional mean of $1_B(c(X_1))$ given $c(X_0)$ under this reference joint law. Hence $T(c(x),B)=P(x,c^{-1}B)$ for $\lambda$-almost every $x$. Intersect over the countable algebra and apply [F5] to obtain equality of these probability measures simultaneously on every Borel $B$, on one $\lambda$-full set. In particular ${\lambda }_{c}T=c_{\ast }(\lambda P)\ll {\lambda }_c$, and ${\pi }_{c}T={\pi }_c$ because $\pi \ll \lambda$ and $\pi P=\pi$.

Iterating that kernel equality gives $T^n(c(x),\cdot )=c_{\ast }{P}^n(x,\cdot )$ for $\lambda$-almost every $x$, simultaneously for all $n\ge 0$. At the induction step, $P(x,\cdot )$ avoids the preceding $\lambda$-null exceptional set for $\lambda$-almost every $x$, since $\lambda P\ll \lambda$; integration of a nonnegative Borel function of $c$ agrees under the two one-step measures by simple approximation and [F8]. The geometric hypothesis therefore implies $\mathrm{\parallel }{T}^n(z,\cdot )-{\pi }_c{\mathrm{\parallel }}_{\mathrm{T}\mathrm{V}}\le m_0(z)r^n$ for ${\lambda }_c$-almost every $z$, simultaneously for $n\ge 1$.

On $C$ the total-variation supremum can be taken over its fixed countable generating algebra. Indeed, for any two probabilities, every Borel set is approximable in symmetric-difference measure by that algebra under their finite sum: the approximable sets form a sigma-field, using complements and first approximating a countable union by a finite union. Its probability difference is approximated too. Thus the supremum over the algebra equals the full supremum and is measurable as a function of the starting point. Let $B_0$ be the Borel set where all the bounds in step 4.1 hold and ${\phi }_0,m_0$ are finite with $m_0\ge 0$. It is ${\lambda }_c$-full by step 2.1 and the finiteness of $f,M$. Define $B_{j+1}=B_j\cap \{z:T(z,B_j)=1\}$ and $B={\bigcap }_{j\ge 0}{B}_j$. Absolute continuity ${\lambda }_{c}T\ll {\lambda }_c$ makes every $B_j$ full. If $z\in B$, continuity from above gives $T(z,B)=1$. Thus $B$ is nonempty, Borel, absorbing and ${\lambda }_c$-full; no Polish topology on $B$ is asserted.

Keep the factor space $S=C$, and define $T_{\ast }(z,\cdot )=T(z,\cdot )$ for $z\in B$ and $T_{\ast }(z,\cdot )={\pi }_c$ otherwise. This is a Borel probability kernel. Set $\widehat{\pi }={\pi }_c$ and $\widehat{\mu }={\mu }_c$; both assign mass one to $B$, since they are absolutely continuous with respect to ${\lambda }_c$. Define $\phi ={\phi }_0$ and $\widehat{M}=m_0$ on $B$, and set both to zero on $C\setminus B$; these functions are finite measurable. Since $B$ is absorbing and ${\pi }_{c}T={\pi }_c$, we have ${\pi }_c{T}_{\ast }={\pi }_c$. For $z\in B$, the iterates of $T_{\ast }$ equal those of $T$, so the geometric bound holds by step 5.1. For $z\mathrm{\notin }B$, $T_{\ast }^n(z,\cdot )={\pi }_c$ for every $n\ge 1$, so it holds with $\widehat{M}(z)=0$. Thus $T_{\ast }$ is the required kernel on the nonempty standard-Borel space $C$.

For an original chain with initial law $\mu$, every $X_k$ has law $\mu P^k\ll \lambda$. Thus step 3.1 and [F6], followed by conditioning on the coded past, show that $c(X_k)$ is a $T$-chain. Since every coded state belongs to $B$ almost surely, this is also a $T_{\ast }$-chain. Countably many time and algebra tests suffice; [F5] and bounded approximation extend them to every bounded Borel test. The code chain remains in $B$ almost surely and $f(X_k)=\phi (c(X_k))$ simultaneously for every $k$. The same argument applies with initial law $\pi$. This proves the complete initial and stationary observation-path equalities and, by pushforward integration, every stated stationary moment identity.

The real observation-path space ${\mathbb{R}}^{{\mathbb{N}}_0}$ is standard Borel: its product topology is Polish by [F9], and its Borel sigma-algebra is the coordinate product sigma-algebra, since the real line has a countable base and finite-coordinate basic open sets form a countable base for this product. Disintegrate any specified joint law of this path and a standard-Borel auxiliary random element conditional on the path by [F4]. Evaluate that kernel at the actual original observation path and extend the original probability space by its kernel integral as in [F7]; explicitly, $P(d\omega )K((f(X_n(\omega )){)}_{n\ge 0},dv)$ is countably additive by [F8] and has marginal $P$. Its observation/auxiliary joint law is exactly the specified law. This proves the lifting assertion without a conditional kernel on the original arbitrary state space.