Uniform ergodicity bounds the uniform-mixing coefficient

Fix $n\ge 1$, $j\ge 0$, $A\in {\mathcal{F}}_j$ with $\mathbb{P}(A)>0$, and $B\in \sigma (X_{j+n},X_{j+n+1},\dots )$. By [F2], there is measurable $h:\mathsf{X}\to [0,1]$ with $\mathbb{E}[{\mathbf{1}}_B\mid {\mathcal{F}}_{j+n}]=h(X_{j+n})$. By [F3] and [F5], $\mathbb{P}(B\mid A)=\mathbb{E}[P^{n}h(X_j)\mid A]$. Stationarity gives $\mathbb{P}(B)=\pi (h)$.

For probability measures $\mu ,\nu$ and measurable $0\le h\le 1$, the dyadic simple functions $h_m=2^{-m}{\sum }_{j=1}^{2^m}{\mathbf{1}}_{\{h\ge j{2}^{-m}\}}$ differ from $h$ by at most $2^{-m}$. Their integrals differ by at most $\mathrm{\parallel }\mu -\nu {\mathrm{\parallel }}_{\mathrm{T}\mathrm{V}}$, since each summand is an indicator and [F4] bounds its measure difference. Letting $m\to \mathrm{\infty }$ gives $\mathrm{\mid }\mu (h)-\nu (h)\mathrm{\mid }\le \mathrm{\parallel }\mu -\nu {\mathrm{\parallel }}_{\mathrm{T}\mathrm{V}}$. Applying this with $\mu =P^n(x,\cdot )$ and $\nu =\pi$ proves $\mathrm{\mid }{P}^{n}h(x)-\pi (h)\mathrm{\mid }\le C{r}^n$ for every $x$.

Averaging the pointwise bound of step 1.2 conditionally on $A$ in step 1.1 gives $\mathrm{\mid }\mathbb{P}(B\mid A)-\mathbb{P}(B)\mathrm{\mid }\le C{r}^n$. Taking the supremum in [F1] proves the result.