Gaussian measure

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In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space $\mathbb{R}^{n}$ , 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.

1 Definitions
2 Properties of Gaussian measure
3 Gaussian measures on infinte-dimensional spaces
4 See also

[edit] Definitions

Let $n \in \mathbb{N}$ and let $\lambda^{n} : \mathrm{Borel} (\mathbb{R}^{n})_{0} \to [0, + \infty]$ denote Lebesgue measure. (Technical point: the subscript "0" indicates that Lebesgue measure is defined on the completion of the Borel sigma algebra.) Then the standard Gaussian measure $\gamma^{n} : \mathrm{Borel} (\mathbb{R}^{n})_{0} \to [0, + \infty]$ is defined by

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

In terms of the Radon-Nikodym derivative,

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

More generally, the Gaussian measure with mean $\mu \in \mathbb{R}^{n}$ and variance $σ 2 > 0$ is given by

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

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 $\sigma \to 0$ , and is considered to be a degenerate Gaussian measure; in contrast, Gaussian measures with finite non-zero variance are called non-degenerate Gaussian measures.

[edit] Properties of Gaussian measure

The standard Gaussian measure $γ n$ 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 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: $\mathrm{supp} (\gamma^{n}) = \mathbb{R}^{n}$ ;
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 $B$ ,

$\gamma^{n} (B) = \sup \{ \gamma^{n} (K) | K \subseteq B \mathrm{\,is\,compact} \},$

so Gaussian measure is a Radon measure;

is not translation-invariant, but does satisfy the relation

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

where the derivative on the left-hand side is the Radon-Nikodym derivative, and $(T h) * (γ n)$ is the push forward of standard Gaussian measure by the translation map $T_{h} : x \mapsto x + h$ ;

is the probability measure associated to a normal probability distribution:

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

[edit] Gaussian measures on infinte-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 the define Gaussian measures on infinte-dimensional spaces, the main example being the abstract Wiener space construction. A Borel measure $μ$ on a separable Banach space $E$ is said to be a non-degenerate Gaussian measure if, for every linear functional $\ell \in E^{*}$ except $\ell = 0$ , the push forward measure $\ell_{*} (\mu)$ is a non-degenerate Gaussian measure on $\mathbb{R}$ in the sense defined above.