Measure-preserving transformation

A probability-measure-preserving transformation of a probability space $(X,\Sigma,\mu)$ is a measurable map $T:X\to X$ such that

\mu(T^{-1}E)=\mu(E)\qquad(E\in\Sigma).

Equivalently, its pushforward satisfies $T_*\mu=\mu$ . Here $T^{-1}E$ means a preimage and does not assume an inverse map.

Invertibility and integration

An invertible transformation has a measurable inverse, possibly after discarding null sets. Measure preservation implies

\int_X f\circ T\,d\mu=\int_X f\,d\mu

for nonnegative measurable or integrable $f$ . This identity makes composition well defined on almost-everywhere equivalence classes.

Examples

Permutations preserve uniform probability on a finite set. The doubling map preserves circle probability without being invertible modulo null sets.