A Gaussian probability space consists of
- a (complete) probability space ,
- a closed linear subspace called the Gaussian space such that all are mean zero Gaussian variables. Their σ-algebra is denoted as .
- a σ-algebra called the transverse σ-algebra which is defined through
- [3]
Subspaces
A subspace of a Gaussian probability space consists of
- a closed subspace ,
- a sub σ-algebra of transverse random variables such that and are independent, and .[3]
Example:
Let be a Gaussian probability space with a closed subspace . Let be the orthogonal complement of in . Since orthogonality implies independence between and , we have that is independent of . Define via .
For we have .
Fundamental algebra
Given a Gaussian probability space one defines the algebra of cylindrical random variables
where is a polynomial in and calls the fundamental algebra. For any it is true that .
For an irreducible Gaussian probability the fundamental algebra is a dense set in for all .[4]