Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Gaussian measure
Summary
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. One reason why Gaussian measures are so ubiquitous in probability theory is the central limit theorem. Loosely speaking, it states that if a random variable X is obtained by summing a large number N of independent random variables of order 1, then X is of order and its law is approximately Gaussian.
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, 1] 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.
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: , where 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, so Gaussian measure is a Radon measure;
is not translation-invariant, but does satisfy the relation where the derivative on the left-hand side is the Radon–Nikodym derivative, and (Th)∗(γn) is the push forward of standard Gaussian measure by the translation map Th : Rn → Rn, Th(x) = x + h;
is the probability measure associated to a normal probability distribution:
It can be shown that there is no analogue of Lebesgue measure on an infinite-dimensional vector space.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Ontological neighbourhood
Related lectures (1)
Related publications (3)
Related concepts (6)
Tightness of measures
In mathematics, tightness is a concept in measure theory. The intuitive idea is that a given collection of measures does not "escape to infinity". Let be a Hausdorff space, and let be a σ-algebra on that contains the topology . (Thus, every open subset of is a measurable set and is at least as fine as the Borel σ-algebra on .) Let be a collection of (possibly signed or complex) measures defined on . The collection is called tight (or sometimes uniformly tight) if, for any , there is a compact subset of such that, for all measures , where is the total variation measure of .
Inner regular measure
In mathematics, an inner regular measure is one for which the measure of a set can be approximated from within by compact subsets. Let (X, T) be a Hausdorff topological space and let Σ be a σ-algebra on X that contains the topology T (so that every open set is a measurable set, and Σ is at least as fine as the Borel σ-algebra on X). Then a measure μ on the measurable space (X, Σ) is called inner regular if, for every set A in Σ, This property is sometimes referred to in words as "approximation from within by compact sets.
Dirac measure
In mathematics, a Dirac measure assigns a size to a set based solely on whether it contains a fixed element x or not. It is one way of formalizing the idea of the Dirac delta function, an important tool in physics and other technical fields. A Dirac measure is a measure δx on a set X (with any σ-algebra of subsets of X) defined for a given x ∈ X and any (measurable) set A ⊆ X by where 1A is the indicator function of A. The Dirac measure is a probability measure, and in terms of probability it represents the almost sure outcome x in the sample space X.