Gaussian measure
Encyclopedia
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
.
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
for any measurable set A ∈ B0(Rn). In terms of the Radon–Nikodym derivative,
More generally, the Gaussian measure with mean
μ ∈ Rn and variance
σ2 > 0 is given by
Gaussian measures with mean μ = 0 are known as centred Gaussian measures.
The Dirac measure δμ is the weak limit of
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.
so Gaussian measure is a Radon 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Gaussian measure is a Borel measure on finite-dimensional Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
Rn, closely related to the normal distribution in statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
. There is also a generalization to infinite-dimensional spaces. Gaussian measures are named after the German
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Carl Friedrich Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
.
Definitions
Let n ∈ N and let B0(Rn) denote the completionComplete measure
In mathematics, a complete measure is a measure space in which every subset of every null set is measurable...
of the Borel σ-algebra on Rn. Let λn : B0(Rn) → [0, +∞] denote the usual n-dimensional Lebesgue measure
Lebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...
. Then the standard Gaussian measure γn : B0(Rn) → [0, +∞] is defined by
for any measurable set A ∈ B0(Rn). In terms of the Radon–Nikodym derivative,
More generally, the Gaussian measure with mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
μ ∈ Rn and variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
σ2 > 0 is given by
Gaussian measures with mean μ = 0 are known as centred Gaussian measures.
The Dirac measure δμ is the weak limit of
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 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 equivalentEquivalence (measure theory)In mathematics, and specifically in measure theory, equivalence is a notion of two measures being "the same".-Definition:Let be a measurable space, and let μ, ν : Σ → R be two signed measures. Then μ is said to be equivalent to ν if and only if each is absolutely continuous with respect to the other...
to Lebesgue measure:
, where 
stands for absolute continuityAbsolute continuityIn mathematics, the relationship between the two central operations of calculus, differentiation and integration, stated by fundamental theorem of calculus in the framework of Riemann integration, is generalized in several directions, using Lebesgue integration and absolute continuity...
of measures; - is supportedSupport (measure theory)In mathematics, the support of a measure μ on a measurable topological space is a precise notion of where in the space X the measure "lives"...
on all of Euclidean space: supp(γn) = Rn; - is a probability measureProbability measureIn mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity...
(γn(Rn) = 1), and so it is locally finiteLocally finite measureIn mathematics, a locally finite measure is a measure for which every point of the measure space has a neighbourhood of finite measure.-Definition:...
; - is strictly positiveStrictly positive measureIn mathematics, strict positivity is a concept in measure theory. Intuitively, a strictly positive measure is one that is "nowhere zero", or that it is zero "only on points".-Definition:...
: every non-empty open setOpen setThe concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
has positive measure; - is inner regularInner regular measureIn mathematics, an inner regular measure is one for which the measure of a set can be approximated from within by compact subsets.-Definition:...
: for all Borel sets A,
so Gaussian measure is a Radon measure
Radon measure
In mathematics , a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space X that is locally finite and inner regular.-Motivation:...
;
- is not translationTranslation (geometry)In Euclidean geometry, a translation moves every point a constant distance in a specified direction. A translation can be described as a rigid motion, other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant vector to every point, or...
-invariantInvariant (mathematics)In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...
, but does satisfy the relation
- where the derivativeDerivativeIn calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
on the left-hand side is the Radon–Nikodym derivative, and (Th)∗(γn) is the push forwardPushforward measureIn measure theory, a pushforward measure is obtained by transferring a measure from one measurable space to another using a measurable function.-Definition:...
of standard Gaussian measure by the translation map Th : Rn → Rn, Th(x) = x + h;- is the probability measure associated to a normal probability distributionProbability distributionIn probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
:
- is the probability measure associated to a normal probability distribution
Gaussian measures on infinite-dimensional spaces
It can be shown that there is no analogue of Lebesgue measureThere is no infinite-dimensional Lebesgue measure
In mathematics, it is a theorem that there is no analogue of Lebesgue measure on an infinite-dimensional Banach space. Other kinds of measures are therefore used on infinite-dimensional spaces: often, the abstract Wiener space construction is used...
on an infinite-dimensional vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
. Even so, it is possible to define Gaussian measures on infinite-dimensional spaces, the main example being the abstract Wiener space
Abstract Wiener space
An abstract Wiener space is a mathematical object in measure theory, used to construct a "decent" measure on an infinite-dimensional vector space. It is named after the American mathematician Norbert Wiener...
construction. A Borel measure γ on a separable Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
E is said to be a non-degenerate (centered) Gaussian measure if, for every linear functional
Linear functional
In linear algebra, a linear functional or linear form is a linear map from a vector space to its field of scalars. In Rn, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product, or the...
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
Classical Wiener space
In mathematics, classical Wiener space is the collection of all continuous functions on a given domain , taking values in a metric space . Classical Wiener space is useful in the study of stochastic processes whose sample paths are continuous functions...
on the space of continuous
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
paths
Path (topology)
In mathematics, a path in a topological space X is a continuous map f from the unit interval I = [0,1] to XThe initial point of the path is f and the terminal point is f. One often speaks of a "path from x to y" where x and y are the initial and terminal points of the path...
is a Gaussian measure.