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.

Contents

Definitions

Properties of Gaussian measure

Gaussian measures on infinite-dimensional spaces

See also

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
:
ight) , mathrm{d} lambda^{n} (x)
for any measurable set ''A'' ∈ ''B''₀('R'^''n''). In terms of the Radon-Nikodym derivative,
:
ight).
More generally, the Gaussian measure with mean ''μ'' ∈ 'R'^''n'' and variance ''σ''² > 0 is given by
:
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
:
★ } (gamma^{n})}{mathrm{d} gamma^{n}} (x) = exp left( langle h, x
angle_{mathbb{R}^{n}} - rac{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.

See also

★ Cameron-Martin theorem