Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 4 / 26J / d

For every $x\in X$, invariance givesThus $T^{-1}\{x\}$ is nonempty, so $T$ is a surjective function. A surjection from a finite set to itself is a bijection. ThereforeThis is the finite full-support measure-preserving map is bijective principle.