Let $(X,\mathcal B,\mu)$ be a probability space, and let $T:X\to X$ be a measurable probability-preserving transformation, meaning $\mu(T^{-1}A)=\mu(A)$ for every $A\in\mathcal B$. Let $\mathcal B_0$ be a [Banach space](/page/Banach%20Space) of $\mu$-[measurable functions](/page/Measurable%20Functions) on $X$, continuously embedded in $L^1(X,\mathcal B,\mu)$, such that $1_X\in \mathcal B_0$. Suppose the transfer operator associated to $T$ preserves $\mathcal B_0$ and defines a [bounded linear operator](/page/Bounded%20Linear%20Operator)

\begin{align*} \mathcal L:\mathcal B_0\to\mathcal B_0 \end{align*}

satisfying, for every $u\in\mathcal B_0$ and every $h\in L^\infty(X,\mathcal B,\mu)$,

\begin{align*} \int_X (\mathcal L u)h\,d\mu(x)=\int_X u(h\circ T)\,d\mu(x). \end{align*}

Assume that $\mathcal L$ has a spectral gap on $\mathcal B_0$ with leading eigenvalue $1$ in the following sense: there exist bounded linear operators

\begin{align*} \Pi:\mathcal B_0\to\mathcal B_0, \qquad N:\mathcal B_0\to\mathcal B_0 \end{align*}

such that

\begin{align*} \mathcal L=\Pi+N, \qquad \Pi N=N\Pi=0, \qquad r(N)<1, \end{align*}

where $r(N)$ denotes the spectral radius of $N$, and

\begin{align*} \Pi u=\left(\int_X u\,d\mu(x)\right)1_X \end{align*}

for every $u\in\mathcal B_0$. Suppose also that there is a constant $C_1>0$ such that

\begin{align*} \|u\|_{L^1(X,\mu)}\le C_1\|u\|_{\mathcal B_0} \end{align*}

for every $u\in\mathcal B_0$.

Then there exist constants $C>0$ and $\rho\in(0,1)$ such that, for every $f\in\mathcal B_0$, every $g\in L^\infty(X,\mathcal B,\mu)$, and every integer $n\ge 0$,

\begin{align*} \left|\int_X f(g\circ T^n)\,d\mu(x)-\left(\int_X f\,d\mu(x)\right)\left(\int_X g\,d\mu(x)\right)\right|\le C\rho^n\|f\|_{\mathcal B_0}\|g\|_{L^\infty(X,\mu)}. \end{align*}