Polish Space

A Polish space is a topological space that is separable and completely metrizable. It is homeomorphic to a complete separable metric space, with no one metric singled out. Polish spaces are the standard base for well-behaved measure theory. Its Borel -algebra is standard Borel, hence Borel-isomorphic to a Borel subset of , and on such a space disintegration, regular conditional distributions, and product measures are all available.

The route to probability is via this standard-Borel structure. For a Polish , the set of Borel probability measures is again Polish under the weak topology, metrized by the Prokhorov metric

where is the -neighbourhood of . Sending and pushing measures forward along Borel maps is the object part of the Giry monad; the unit is the Dirac embedding and the multiplication integrates a measure-on-measures. Because stays Polish, the monad remains inside the same category it started in. Every smooth manifold is Polish, so this machinery applies to state spaces.

Definition. Standard Borel space

A measurable space is standard Borel if it is the Borel structure of some Polish topology. The choice of compatible metric is forgotten; only the Borel sets are retained. This is the exact amount of structure the Giry monad needs, since measures see the -algebra, not the metric.

Remark. Why pass through Polish spaces at all

The forgetful functor from metric spaces (with, say, continuous maps) to topological spaces has an image subcategory, the metrizable spaces, but admits no left adjoint / free functor. Metrizability says a compatible metric exists without fixing a canonical one, and there is no natural way to choose one, so the “free metric on a topological space” does not exist. Analysis needs an actual metric for completeness, Prokhorov, and tightness, whereas probability needs only the Borel structure. One therefore does not try to functorially metrize ; one instead works in the well-behaved slice of Polish and standard-Borel spaces, where a compatible complete metric is available and the Borel structure is canonical.