Gaussian measure

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.

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, +∞] is defined by

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

for any measurable set "A" ∈ "B"0(R"n"). In terms of the Radon-Nikodym derivative,

: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} ight).

More generally, the Gaussian measure with mean "μ" ∈ 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} ight) , 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 "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""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: lambda^{n} ll gamma^{n} ll lambda^{n}, where ll 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",

:gamma^{n} (A) = sup { gamma^{n} (K) | K subseteq A, K mbox{ 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 angle_{mathbb{R}^{n - frac{1}{2} | h |_{mathbb{R}^{2^{2} ight),

: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:

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

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 infinte-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 (centred) Gaussian measure if, for every linear functional "L" ∈ "E"∗ except "L" = 0, the push-forward measure "L"∗("γ") is a non-degenerate (centred) Gaussian measure on R in the sense defined above.

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

ee also

* Cameron-Martin theorem


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