Finite full-support measure-preserving map is bijective
= Finite full-support measure-preserving map is bijective
Let $X$ be a <finite set>, let every singleton have positive probability, and let $T:X\to X$ preserve that probability measure. For every $x\in X$,
$$
\mu(T^{-1}\{x\})=\mu(\{x\})>0,
$$
so every $x$ has a preimage. Thus $T$ is a <surjection>, and every surjection from a finite set to itself is a <bijection>.