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 denote the completion of the Borel \sigma-algebra on. Let denote the usual n-dimensional Lebesgue measure. Then the standard Gaussian measure is defined by for any measurable set. In terms of the Radon–Nikodym derivative, More generally, the Gaussian measure with mean and variance is given by Gaussian measures with mean \mu = 0 are known as centered Gaussian measures. The Dirac measure \delta_\mu is the weak limit of as, 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

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.