Gaussian measure

In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space $R^{n}$ , closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are named after the German mathematician Carl Friedrich Gauss. One reason why Gaussian measures are so ubiquitous in probability theory is the central limit theorem. Loosely speaking, it states that if a random variable $X$ is obtained by summing a large number $N$ of independent random variables with variance 1, then $X$ has variance $N$ and its law is approximately Gaussian.

Definitions

Let $n\in N$ and let $B_{0}(\mathbb {R} ^{n})$ denote the completion of the Borel $\sigma$ -algebra on $\mathbb {R} ^{n}$ . Let $\lambda ^{n}:B_{0}(\mathbb {R} ^{n})\to [0,+\infty ]$ denote the usual $n$ -dimensional Lebesgue measure. Then the standard Gaussian measure $\gamma ^{n}:B_{0}(\mathbb {R} ^{n})\to [0,1]$ is defined by $\gamma ^{n}(A)={\frac {1}{{\sqrt {2\pi }}^{n}}}\int _{A}\exp \left(-{\frac {1}{2}}\left\|x\right\|_{\mathbb {R} ^{n}}^{2}\right)\,\mathrm {d} \lambda ^{n}(x)$ for any measurable set $A\in B_{0}(\mathbb {R} ^{n})$ . In terms of the Radon–Nikodym derivative, ${\frac {\mathrm {d} \gamma ^{n}}{\mathrm {d} \lambda ^{n}}}(x)={\frac {1}{{\sqrt {2\pi }}^{n}}}\exp \left(-{\frac {1}{2}}\left\|x\right\|_{\mathbb {R} ^{n}}^{2}\right).$

More generally, the Gaussian measure with mean $\mu \in \mathbb {R} ^{n}$ and variance $\sigma ^{2}>0$ is given by $\gamma _{\mu ,\sigma ^{2}}^{n}(A):={\frac {1}{{\sqrt {2\pi \sigma ^{2}}}^{n}}}\int _{A}\exp \left(-{\frac {1}{2\sigma ^{2}}}\left\|x-\mu \right\|_{\mathbb {R} ^{n}}^{2}\right)\,\mathrm {d} \lambda ^{n}(x).$

Gaussian measures with mean $\mu =0$ are known as centered Gaussian measures.

The Dirac measure $\delta _{\mu }$ is the weak limit of $\gamma _{\mu ,\sigma ^{2}}^{n}$ as $\sigma \to 0$ , and is considered to be a degenerate Gaussian measure; in contrast, Gaussian measures with finite, non-zero variance are called non-degenerate Gaussian measures.

Properties

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

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: $\lambda ^{n}\ll \gamma ^{n}\ll \lambda ^{n}$ , where $\ll$ stands for absolute continuity of measures;
is supported on all of Euclidean space: $\operatorname {supp} (\gamma ^{n})=\mathbb {R} ^{n}$ ;
is a probability measure $(\gamma ^{n}(\mathbb {R} ^{n})=1)$ , and so it is locally finite;
is strictly positive: every non-empty open set has positive measure;
is inner regular: for all Borel sets $A$ , $\gamma ^{n}(A)=\sup\{\gamma ^{n}(K)\mid K\subseteq A,K{\text{ is compact}}\},$ so Gaussian measure is a Radon measure;
is not translation-invariant, but does satisfy the relation ${\frac {\mathrm {d} (T_{h})_{*}(\gamma ^{n})}{\mathrm {d} \gamma ^{n}}}(x)=\exp \left(\langle h,x\rangle _{\mathbb {R} ^{n}}-{\frac {1}{2}}\|h\|_{\mathbb {R} ^{n}}^{2}\right),$ where the derivative on the left-hand side is the Radon–Nikodym derivative, and $(T_{h})_{*}(\gamma ^{n})$ is the push forward of standard Gaussian measure by the translation map $T_{h}:\mathbb {R} ^{n}\to \mathbb {R} ^{n}$ , $T_{h}(x)=x+h$ ;
is the probability measure associated to a normal probability distribution: $Z\sim \operatorname {Normal} (\mu ,\sigma ^{2})\implies \mathbb {P} (Z\in A)=\gamma _{\mu ,\sigma ^{2}}^{n}(A).$

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 $\gamma$ on a separable Banach space $E$ is said to be a non-degenerate (centered) Gaussian measure if, for every linear functional $L\in E^{*}$ except $L=0$ , the push-forward measure $L_{*}(\gamma )$ is a non-degenerate (centered) Gaussian measure on $\mathbb {R}$ in the sense defined above.

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

References

Bogachev, Vladimir (1998). Gaussian Measures. American Mathematical Society. ISBN 978-1470418694.
Stroock, Daniel (2010). "Gaussian Measures on a Banach Space". Probability Theory: An Analytic View. Cambridge University Press. pp. 264–303. doi:10.1017/9781009549035.010. ISBN 978-0521132503.

Analysis in topological vector spaces
Basic concepts	Abstract Wiener space Classical Wiener space Bochner space Convex series Cylinder set measure Infinite-dimensional vector function Matrix calculus Vector calculus
Derivatives	Differentiable vector–valued functions from Euclidean space Differentiation in Fréchet spaces Fréchet derivative Total Functional derivative Gateaux derivative Directional Generalizations of the derivative Hadamard derivative Holomorphic Quasi-derivative
Measurability	Besov measure Cylinder set measure Canonical Gaussian Classical Wiener measure Measure like set functions infinite-dimensional Gaussian measure Projection-valued Vector Bochner / Weakly / Strongly measurable function Radonifying function
Integrals	Bochner Direct integral Dunford Gelfand–Pettis/Weak Regulated Paley–Wiener
Results	Cameron–Martin theorem Inverse function theorem Nash–Moser theorem Feldman–Hájek theorem No infinite-dimensional Lebesgue measure Sazonov's theorem Structure theorem for Gaussian measures
Related	Crinkled arc Covariance operator
Functional calculus	Borel functional calculus Continuous functional calculus Holomorphic functional calculus
Applications	Banach manifold (bundle) Convenient vector space Choquet theory Fréchet manifold Hilbert manifold

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