Gaussian measure
In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space R, 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 of order 1, then "X" is of order formula_1 and its law is approximately Gaussian. Definitions. Let "n  ∈ N and let B"(R) denote the completion of the Borel "σ"-algebra on R. Let "λ" : "B"(R) → [0, +∞] denote the usual "n"-dimensional Lebesgue measure. Then the standard Gaussian measure "γ" : "B"(R) → [0, 1] is defined by for any measurable set "A" ∈ "B"(R). In terms of the Radon–Nikodym derivative, More generally, the Gaussian measure with mean "μ" ∈ R and variance "σ" > 0 is given by Gaussian measures with mean "μ" = 0 are known as centred Gaussian measures. The Dirac measure "δ" is the weak limit of formula_5 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" on R 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 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.