Automorphism group of a probability space

The automorphism group $\operatorname{Aut}(X,\mu)$ consists of measure-space isomorphisms from a probability space to itself, modulo almost-everywhere equality. Its operation is composition, its identity is the identity map, and inversion is the measurable inverse modulo null sets.

Endomorphisms

All measure-preserving self-maps, including noninvertible ones, form a monoid instead. Ergodic automorphisms generally do not form a subgroup: on a nontrivial space, the identity is not ergodic, while $T^{-1}$ is ergodic whenever $T$ is.

Finite and Gaussian models

For a uniform $n$ -point space this group is $S_n$ . For an atomless standard space it is isomorphic, by conjugation, to the automorphism group of Lebesgue probability on an interval; see the isomorphism theorem.