Definition
Standard probability space
A probability space modeled modulo null sets by a completed Borel probability measure on a Polish space.
A standard probability space is a probability space that, modulo null sets, is the completion of a Borel probability measure on a standard Borel space.
Why the hypothesis appears
This regularity permits concrete measurable models, disintegration, and classification statements such as ergodic decomposition and the Halmos–von Neumann theorem. It is not a hypothesis in the elementary fixed-function criterion for ergodicity.