Gaussian measure

Let n \in N and let B_0(\mathbb{R}^n) denote the completion of the Borel \sigma-algebra on \mathbb{R}^n. Let \lambda^n : B_0(\mathbb{R}^n) \to [0, +\infty] denote the usual n-dimensional Lebesgue measure. Then the standard Gaussian measure \gamma^n : B_0(\mathbb{R}^n) \to [0, 1] is defined by \gamma^{n} (A) = \frac{1}{\sqrt{2 \pi}^{n}} \int_{A} \exp \left( - \frac{1}{2} \left\| x \right\|_{\mathbb{R}^{n}}^{2} \right) \, \mathrm{d} \lambda^{n} (x) for any measurable set A \in B_0(\mathbb{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} \left\| x \right\|_{\mathbb{R}^{n}}^{2} \right). More generally, the Gaussian measure with mean \mu \in \mathbb{R}^n and variance \sigma^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}} \left\| x - \mu \right\|_{\mathbb{R}^{n}}^{2} \right) \, \mathrm{d} \lambda^{n} (x). Gaussian measures with mean \mu = 0 are known as centered Gaussian measures. The Dirac measure \delta_\mu is the weak limit of \gamma_{\mu, \sigma^{2}}^{n} as \sigma \to 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==

The standard Gaussian measure \gamma^n on \mathbb{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: \operatorname{supp}(\gamma^n) = \mathbb{R}^n; • is a probability measure (\gamma^n(\mathbb{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) \mid K \subseteq A, K \text{ 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_{\R^n} - \frac{1}{2} \| h \|_{\R^n}^2 \right), where the derivative on the left-hand side is the Radon–Nikodym derivative, and (T_h)_*(\gamma^n) is the push forward of standard Gaussian measure by the translation map T_h : \mathbb{R}^n \to \mathbb{R}^n, T_h(x) = x + h; • is the probability measure associated to a normal probability distribution: Z \sim \operatorname{Normal} (\mu, \sigma^2) \implies \mathbb{P} (Z \in A) = \gamma_{\mu, \sigma^2}^n (A). ==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. ==See also==