Gaussian measure

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In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space Rⁿ, 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.

1 Definitions
2 Properties of Gaussian measure
3 Gaussian measures on infinite-dimensional spaces
4 See also

[edit] Definitions

Let n ∈ N and let B₀(Rⁿ) denote the completion of the Borel σ-algebra on Rⁿ. Let λⁿ : B₀(Rⁿ) → [0, +∞] denote the usual n-dimensional Lebesgue measure. Then the standard Gaussian measure γⁿ : B₀(Rⁿ) → [0, +∞] is defined by

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

for any measurable set A ∈ B₀(Rⁿ). In terms of the Radon-Nikodym derivative,

$\frac{\mathrm{d} \gamma^{n}}{\mathrm{d} \lambda^{n}} (x) = \exp \left( - \frac{1}{2} \| x \|_{\mathbb{R}^{n}}^{2} \right).$

More generally, the Gaussian measure with mean μ ∈ Rⁿ and variance σ² > 0 is given by

$\gamma_{\mu, \sigma^{2}}^{n} (A) := \frac{1}{\sqrt{2 \pi \sigma^{2}}^{n}} \int_{A} \exp \left( - \frac{1}{2 \sigma^{2}} \| x - \mu \|_{\mathbb{R}^{n}}^{2} \right) \, \mathrm{d} \lambda^{n} (x).$

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.

[edit] Properties of Gaussian measure

The standard Gaussian measure γⁿ on Rⁿ

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(γⁿ) = Rⁿ;
is a probability measure (γⁿ(Rⁿ) = 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

$\frac{\mathrm{d} (T_{h})_{*} (\gamma^{n})}{\mathrm{d} \gamma^{n}} (x) = \exp \left( \langle h, x \rangle_{\mathbb{R}^{n}} - \frac{1}{2} \| h \|_{\mathbb{R}^{2}}^{2} \right),$

where the derivative on the left-hand side is the Radon-Nikodym derivative, and (T_h)_∗(γⁿ) is the push forward of standard Gaussian measure by the translation map T_h : Rⁿ → Rⁿ, T_h(x) = x + h;

is the probability measure associated to a normal probability distribution:

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

[edit] 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.