Gaussian measure

Properties

The standard Gaussian measure ${\displaystyle {\gamma }^n}$ on ${\displaystyle {\mathbb{R}}^n}$

• is a Borel measure (in fact, as remarked above, it is defined on the completion of the Borel sigma algebra, which is a finer structure);
• is equivalent to Lebesgue measure: ${\displaystyle {\lambda }^n\ll {\gamma }^n\ll {\lambda }^n}$, where ${\displaystyle \ll }$ stands for absolute continuity of measures;
• is supported on all of Euclidean space: ${\displaystyle \mathrm{supp}({\gamma }^n)={\mathbb{R}}^n}$;
• is a probability measure ${\displaystyle ({\gamma }^n({\mathbb{R}}^n)=1)}$, and so it is locally finite;
• is strictly positive: every non-empty open set has positive measure;
• is inner regular: for all Borel sets ${\displaystyle A}$, $${\displaystyle {\gamma }^n(A)=sup\{{\gamma }^n(K)\mid K\subseteq A,K\text{ is compact}\},}$$ so Gaussian measure is a Radon measure;
• is not translation-invariant, but does satisfy the relation $${\displaystyle \frac{\mathrm{d}(T_h{)}_{\ast }({\gamma }^n)}{\mathrm{d}{\gamma }^n}(x)=\exp \left(⟨h,x{⟩}_{{\mathbb{R}}^n}-\frac{1}{2}\Vert h{\Vert }_{{\mathbb{R}}^n}^2\right),}$$ where the derivative on the left-hand side is the Radon–Nikodym derivative, and ${\displaystyle (T_h{)}_{\ast }({\gamma }^n)}$ is the push forward of standard Gaussian measure by the translation map ${\displaystyle T_h:{\mathbb{R}}^n\to {\mathbb{R}}^n}$, ${\displaystyle T_h(x)=x+h}$;
• is the probability measure associated to a normal probability distribution: $${\displaystyle Z\sim \mathrm{Normal}(\mu ,{\sigma }^2)⟹\mathbb{P}(Z\in A)={\gamma }_{\mu ,{\sigma }^2}^n(A).}$$

Infinite-dimensional spaces

It can be shown that there is no analogue of Lebesgue measure on an infinite-dimensional vector space. Even so, it is possible to define Gaussian measures on infinite-dimensional spaces, the main example being the abstract Wiener space construction. A Borel measure ${\displaystyle \gamma }$ on a separable Banach space ${\displaystyle E}$ is said to be a non-degenerate (centered) Gaussian measure if, for every linear functional ${\displaystyle L\in E^{\ast }}$ except ${\displaystyle L=0}$, the push-forward measure ${\displaystyle L_{\ast }(\gamma )}$ is a non-degenerate (centered) Gaussian measure on ${\displaystyle \mathbb{R}}$ in the sense defined above.

For example, classical Wiener measure on the space of continuous paths is a Gaussian measure.