Let $(E,\mathcal E)$ be a measurable space, let $\mu$ be a probability measure on $(E,\mathcal E)$, and let $T:E\to E$ be an $\mathcal E/\mathcal E$-measurable map. Let $\mathcal C$ be a class of bounded $\mathcal E/\mathcal B(\mathbb R)$-[measurable functions](/page/Measurable%20Functions) $f:E\to\mathbb R$ with the following measure-determining property: for any probability measures $\rho$ and $\eta$ on $(E,\mathcal E)$, if

\begin{align*} \int_E f\,d\rho(x)=\int_E f\,d\eta(x) \end{align*}

for every $f\in\mathcal C$, then $\rho=\eta$.

If

\begin{align*} \int_E f\circ T\,d\mu(x)=\int_E f\,d\mu(x) \end{align*}

for every $f\in\mathcal C$, then $T$ is probability-preserving; equivalently,

\begin{align*} \mu(T^{-1}(A))=\mu(A) \end{align*}

for every $A\in\mathcal E$.