Factor of a probability-preserving system

A factor map between probability-preserving systems $(X,\mu,T)$ and $(Y,\nu,S)$ is a measurable map $p:X\to Y$ such that

p_*\mu=\nu,\qquad p\circ T=S\circ p\quad\text{almost everywhere}.

The target system is a factor of the source. For a group action the intertwining identity is required for every group element.

Observable interpretation

Pullback $f\mapsto f\circ p$ embeds $L^2(Y,\nu)$ isometrically into $L^2(X,\mu)$ as an invariant space of observables. On standard spaces, factors can equivalently be described by invariant sub-sigma-algebras modulo null sets.

An arbitrary invariant Hilbert subspace need not define a factor: the corresponding bounded observables must also retain multiplication and complex conjugation.