In mathematics, a Gaussian measure is a Borel measure on finite-dimensional Euclidean space, 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 and its law is approximately Gaussian.

Definitions

denote the completion of the \sigma-algebra-dimensional Lebesgue measure. Then the standard Gaussian measure. In terms of the Radon–Nikodym derivative

More generally, the Gaussian measure with mean and variance

are known as centered Gaussian measures.

The Dirac measure is the weak limit, 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

• 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 stands for absolute continuity of measures;

• is supported;

• is a probability measure, and so it is locally finite;

• is strictly positive: every non-empty open set has positive measure;

• is inner regular so Gaussian measure is a Radon measure;

• is not translation-invariant where the derivative on the left-hand side is the Radon–Nikodym derivative is the push forward;

• is the probability measure associated to a normal probability distribution

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 on a separable Banach space is said to be a non-degenerate (centered) Gaussian measure if, for every linear functional, the push-forward measure in the sense defined above.

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

See also

References