# 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.

Definitions

Let "n" &isin; N and let "B"0(R"n") denote the completion of the Borel "&sigma;"-algebra on R"n". Let "&lambda;""n" : "B"0(R"n") &rarr; [0, +&infin;] denote the usual "n"-dimensional Lebesgue measure. Then the standard Gaussian measure "&gamma;""n" : "B"0(R"n") &rarr; [0, +&infin;] is defined by

:$gamma^\left\{n\right\} \left(A\right) = frac\left\{1\right\}\left\{sqrt\left\{2 pi\right\}^\left\{n int_\left\{A\right\} exp left\left( - frac\left\{1\right\}\left\{2\right\} | x |_\left\{mathbb\left\{R\right\}^\left\{n^\left\{2\right\} ight\right) , mathrm\left\{d\right\} lambda^\left\{n\right\} \left(x\right)$

for any measurable set "A" &isin; "B"0(R"n"). In terms of the Radon-Nikodym derivative,

:$frac\left\{mathrm\left\{d\right\} gamma^\left\{n\left\{mathrm\left\{d\right\} lambda^\left\{n \left(x\right) = frac\left\{1\right\}\left\{sqrt\left\{2 pi\right\}^\left\{n exp left\left( - frac\left\{1\right\}\left\{2\right\} | x |_\left\{mathbb\left\{R\right\}^\left\{n^\left\{2\right\} ight\right).$

More generally, the Gaussian measure with mean "&mu;" &isin; R"n" and variance "&sigma;"2 &gt; 0 is given by

:$gamma_\left\{mu, sigma^\left\{2^\left\{n\right\} \left(A\right) := frac\left\{1\right\}\left\{sqrt\left\{2 pi sigma^\left\{2^\left\{n int_\left\{A\right\} exp left\left( - frac\left\{1\right\}\left\{2 sigma^\left\{2 | x - mu |_\left\{mathbb\left\{R\right\}^\left\{n^\left\{2\right\} ight\right) , mathrm\left\{d\right\} lambda^\left\{n\right\} \left(x\right).$

Gaussian measures with mean "&mu;" = 0 are known as centred Gaussian measures.

The Dirac measure "&delta;""&mu;" is the weak limit of $gamma_\left\{mu, sigma^\left\{2^\left\{n\right\}$ as "&sigma;" &rarr; 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 of Gaussian measure

The standard Gaussian measure "&gamma;""n" on 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: $lambda^\left\{n\right\} ll gamma^\left\{n\right\} ll lambda^\left\{n\right\}$, where $ll$ stands for absolute continuity of measures;
* is supported on all of Euclidean space: supp("&gamma;""n") = R"n";
* is a probability measure ("&gamma;""n"(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",

:$gamma^\left\{n\right\} \left(A\right) = sup \left\{ gamma^\left\{n\right\} \left(K\right) | K subseteq A, K mbox\left\{ is compact\right\} \right\},$

so Gaussian measure is a Radon measure;
* is not translation-invariant, but does satisfy the relation

:$frac\left\{mathrm\left\{d\right\} \left(T_\left\{h\right\}\right)_\left\{*\right\} \left(gamma^\left\{n\right\}\right)\right\}\left\{mathrm\left\{d\right\} gamma^\left\{n \left(x\right) = exp left\left( langle h, x angle_\left\{mathbb\left\{R\right\}^\left\{n - frac\left\{1\right\}\left\{2\right\} | h |_\left\{mathbb\left\{R\right\}^\left\{2^\left\{2\right\} ight\right),$

:where the derivative on the left-hand side is the Radon-Nikodym derivative, and ("T""h")&lowast;("&gamma;""n") is the push forward of standard Gaussian measure by the translation map "T""h" : R"n" &rarr; R"n", "T""h"("x") = "x" + "h";
* is the probability measure associated to a normal probability distribution:

:$Z sim mathrm\left\{Normal\right\} \left(mu, sigma^\left\{2\right\}\right) implies mathbb\left\{P\right\} \left(Z in A\right) = gamma_\left\{mu, sigma^\left\{2^\left\{n\right\} \left(A\right).$

Gaussian measures on 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 infinte-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 (centred) Gaussian measure if, for every linear functional "L" &isin; "E"&lowast; except "L" = 0, the push-forward measure "L"&lowast;("&gamma;") is a non-degenerate (centred) Gaussian measure on R in the sense defined above.

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

ee also

* Cameron-Martin theorem

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