Definition
Standard Borel space
A measurable space isomorphic to the Borel space of a Polish space.
A standard Borel space is a measurable space for which there are a Polish space and a bijection such that both and are measurable, with carrying its Borel sigma-algebra.
Examples
Finite and countable discrete spaces, Euclidean spaces, compact metric spaces, and their Borel subsets are standard Borel. The definition specifies measurable structure; it does not choose a preferred topology on .
Completion
Completing a Borel probability measure adds subsets of null sets. The completed measurable space need not itself be standard Borel. The related notion of a standard probability space allows this completion and identification modulo null sets.