Isomorphism theorem for atomless standard probability spaces

Statement

Every atomless standard probability space is isomorphic modulo null sets to $([0,1],\mathcal L,\lambda)$ , where $\lambda$ is Lebesgue probability and $\mathcal L$ its completed sigma-algebra.

Consequence for dynamics

Consequently, an invertible probability-preserving system on any such space can be transported to any other by $T\mapsto bTb^{-1}$ . This preserves ergodicity and mixing. In particular, changing the finite dimension of a Gaussian probability space imposes no restriction on its measurable dynamics up to conjugacy.

Scope

The isomorphism is measure theoretic. A square and an interval satisfy the theorem even though they are not homeomorphic.

References

Jesse Peterson, Lecture notes on ergodic theory, 2011. §2.3, “A remark about measure spaces.”