# Gaussian measure

In

mathematics ,**Gaussian measure**is aBorel measure on finite-dimensionalEuclidean space **R**^{"n"}, closely related to thenormal distribution instatistics . There is also a generalization to infinite-dimensional spaces. Gaussian measures are named after the Germanmathematician Carl Friedrich Gauss .**Definitions**Let "n" ∈

**N**and let "B"_{0}(**R**^{"n"}) denote the completion of the Borel "σ"-algebra on**R**^{"n"}. Let "λ"^{"n"}: "B"_{0}(**R**^{"n"}) → [0, +∞] denote the usual "n"-dimensionalLebesgue measure . Then the**standard Gaussian measure**"γ"^{"n"}: "B"_{0}(**R**^{"n"}) → [0, +∞] is defined by:$gamma^\{n\}\; (A)\; =\; frac\{1\}\{sqrt\{2\; pi\}^\{n\; int\_\{A\}\; exp\; left(\; -\; frac\{1\}\{2\}\; |\; x\; |\_\{mathbb\{R\}^\{n^\{2\}\; ight)\; ,\; mathrm\{d\}\; lambda^\{n\}\; (x)$

for any measurable set "A" ∈ "B"

_{0}(**R**^{"n"}). In terms of theRadon-Nikodym derivative ,:$frac\{mathrm\{d\}\; gamma^\{n\{mathrm\{d\}\; lambda^\{n\; (x)\; =\; frac\{1\}\{sqrt\{2\; pi\}^\{n\; exp\; left(\; -\; frac\{1\}\{2\}\; |\; x\; |\_\{mathbb\{R\}^\{n^\{2\}\; ight).$

More generally, the Gaussian measure with

mean "μ" ∈**R**^{"n"}andvariance "σ"^{2}> 0 is given by:$gamma\_\{mu,\; sigma^\{2^\{n\}\; (A)\; :=\; frac\{1\}\{sqrt\{2\; pi\; sigma^\{2^\{n\; int\_\{A\}\; exp\; left(\; -\; frac\{1\}\{2\; sigma^\{2\; |\; x\; -\; mu\; |\_\{mathbb\{R\}^\{n^\{2\}\; ight)\; ,\; mathrm\{d\}\; lambda^\{n\}\; (x).$

Gaussian measures with mean "μ" = 0 are known as

**centred Gaussian measures**.The

Dirac measure "δ"_{"μ"}is the weak limit of $gamma\_\{mu,\; sigma^\{2^\{n\}$ as "σ" → 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 "γ"

^{"n"}on**R**^{"n"}

* is aBorel 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 forabsolute continuity of measures;

* is supported on all of Euclidean space: supp("γ"^{"n"}) =**R**^{"n"};

* is aprobability measure ("γ"^{"n"}(**R**^{"n"}) = 1), and so it is locally finite;

* is strictly positive: every non-emptyopen set has positive measure;

* is inner regular: for all Borel sets "A",:$gamma^\{n\}\; (A)\; =\; sup\; \{\; gamma^\{n\}\; (K)\; |\; K\; subseteq\; A,\; K\; mbox\{\; is\; compact\}\; \},$

so Gaussian measure is a

Radon measure ;

* is not translation-invariant, but does satisfy the relation:$frac\{mathrm\{d\}\; (T\_\{h\})\_\{*\}\; (gamma^\{n\})\}\{mathrm\{d\}\; gamma^\{n\; (x)\; =\; exp\; left(\; langle\; h,\; x\; angle\_\{mathbb\{R\}^\{n\; -\; frac\{1\}\{2\}\; |\; h\; |\_\{mathbb\{R\}^\{2^\{2\}\; ight),$

:where the

derivative on the left-hand side is theRadon-Nikodym derivative , and ("T"_{"h"})_{∗}("γ"^{"n"}) is the push forward of standard Gaussian measure by the translation map "T"_{"h"}:**R**^{"n"}→**R**^{"n"}, "T"_{"h"}("x") = "x" + "h";

* is the probability measure associated to a normalprobability distribution ::$Z\; sim\; mathrm\{Normal\}\; (mu,\; sigma^\{2\})\; implies\; mathbb\{P\}\; (Z\; in\; A)\; =\; gamma\_\{mu,\; sigma^\{2^\{n\}\; (A).$

**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 theabstract Wiener space construction. A Borel measure "γ" on aseparable Banach space "E" is said to be a**non-degenerate (centred) Gaussian measure**if, for everylinear functional "L" ∈ "E"^{∗}except "L" = 0, the push-forward measure "L"_{∗}("γ") 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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