Standard Probability Space - Verifying The Standardness

Verifying The Standardness

Every probability distribution on the space turns it into a standard probability space. (Here, a probability distribution means a probability measure defined initially on the Borel sigma-algebra and completed.)

The same holds on every Polish space, see (Rokhlin 1962, Sect. 2.7 (p. 24)), (Haezendonck 1973, Example 1 (p. 248)), (de la Rue 1993, Theorem 2-3), and (Itô 1984, Theorem 2.4.1).

For example, the Wiener measure turns the Polish space (of all continuous functions endowed with the topology of local uniform convergence) into a standard probability space.

Another example: for every sequence of random variables, their joint distribution turns the Polish space (of sequences; endowed with the product topology) into a standard probability space.

(Thus, the idea of dimension, very natural for topological spaces, is utterly inappropriate for standard probability spaces.)

The product of two standard probability spaces is a standard probability space.

The same holds for the product of countably many spaces, see (Rokhlin 1962, Sect. 3.4), (Haezendonck 1973, Proposition 12), and (Itô 1984, Theorem 2.4.3).

A measurable subset of a standard probability space is a standard probability space. It is assumed that the set is not a null set, and is endowed with the conditional measure. See (Rokhlin 1962, Sect. 2.3 (p. 14)) and (Haezendonck 1973, Proposition 5).

Every probability measure on a standard Borel space turns it into a standard probability space.