# Gaussian measure

__: Type of Borel measure__

**Short description**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 *Germany* 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 of order 1, then *X* is of order [math]\displaystyle{ \sqrt{N} }[/math] and its law is approximately Gaussian.

## 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*-dimensional Lebesgue measure. Then the **standard Gaussian measure** *γ*^{n} : *B*_{0}(**R**^{n}) → [0, 1] is defined by

- [math]\displaystyle{ \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) }[/math]

for any measurable set *A* ∈ *B*_{0}(**R**^{n}). In terms of the Radon–Nikodym derivative,

- [math]\displaystyle{ \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} \right). }[/math]

More generally, the Gaussian measure with mean *μ* ∈ **R**^{n} and variance *σ*^{2} > 0 is given by

- [math]\displaystyle{ \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). }[/math]

Gaussian measures with mean *μ* = 0 are known as **centred Gaussian measures**.

The Dirac measure *δ*_{μ} is the weak limit of [math]\displaystyle{ \gamma_{\mu, \sigma^{2}}^{n} }[/math] 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

The standard Gaussian measure *γ*^{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: [math]\displaystyle{ \lambda^{n} \ll \gamma^n \ll \lambda^n }[/math], where [math]\displaystyle{ \ll }[/math] stands for absolute continuity of measures;
- is supported on all of Euclidean space: supp(
*γ*^{n}) =**R**^{n}; - is a probability measure (
*γ*^{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*, [math]\displaystyle{ \gamma^n (A) = \sup \{ \gamma^n (K) \mid K \subseteq A, K \text{ is compact} \}, }[/math] so Gaussian measure is a Radon measure; - is not translation-invariant, but does satisfy the relation [math]\displaystyle{ \frac{\mathrm{d} (T_h)_{*} (\gamma^n)}{\mathrm{d} \gamma^n} (x) = \exp \left( \langle h, x \rangle_{\R^n} - \frac{1}{2} \| h \|_{\R^n}^2 \right), }[/math] 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}:**R**^{n}→**R**^{n},*T*_{h}(*x*) =*x*+*h*; - is the probability measure associated to a normal probability distribution: [math]\displaystyle{ Z \sim \operatorname{Normal} (\mu, \sigma^2) \implies \mathbb{P} (Z \in A) = \gamma_{\mu, \sigma^2}^n (A). }[/math]

## 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 *γ* on a separable Banach space *E* is said to be a **non-degenerate (centered) Gaussian measure** if, for every linear functional *L* ∈ *E*^{∗} except *L* = 0, the push-forward measure *L*_{∗}(*γ*) is a non-degenerate (centered) Gaussian measure on **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).
*Probability Theory: An Analytic View*. Cambridge University Press. ISBN 978-0521132503.

## See also

- Besov measure - a generalisation of Gaussian measure
- Cameron–Martin theorem – Theorem of measure theory
- Covariance operator – Operator in probability theory
- Feldman–Hájek theorem

Original source: https://en.wikipedia.org/wiki/Gaussian measure.
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