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In mathematics, specifically in measure theory, the trivial measure on any measurable space (X, Σ) is the measure μ which assigns zero measure to every measurable set: μ(A) = 0 for all A in Σ.
Let μ denote the trivial measure on some measurable space (X, Σ).
Suppose that X is a topological space and that Σ is the Borel σ-algebra on X.
In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets. Some authors require additional restrictions on the measure, as described below.
In measure theory, a branch of mathematics, 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 n-dimensional volume, n-volume, or simply volume. It is used throughout real analysis, in particular to define Lebesgue integration. Sets that can be assigned a Lebesgue measure are called Lebesgue-measurable; the measure of the Lebesgue-measurable set A is here denoted by λ(A).
In mathematics, a measure on a set is a systematic way to assign a number, intuitively interpreted as its size, to some subsets of that set, called measurable sets. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the usual length, area, or volume to subsets of a Euclidean spaces, for which this be defined. For instance, the Lebesgue measure of an interval of real numbers is its usual length.
In mathematical analysis and in probability theory, a σ-algebra on a set X is a collection Σ of subsets of X that includes X itself, is closed under complement, and is closed under countable unions.
In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.
In mathematics, a complete measure is a measure space in which every subset of every null set is measurable. More formally, a measure space (X, Σ, μ) is complete if and only if
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue, although according to the Bourbaki group they were first introduced by Frigyes Riesz. Lp spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, finance, engineering, and other disciplines.
In mathematics, the Radon–Nikodym theorem is a result in measure theory that expresses the relationship between two measures defined on the same measurable space. A measure is a set function that assigns a consistent magnitude to the measurable subsets of a measurable space. Examples of a measure include area and volume, where the subsets are sets of points; or the probability of an event, which is a subset of possible outcomes within a wider probability space.
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 finite on all compact sets, outer regular on all Borel sets, and inner regular on open sets. These conditions guarantee that the measure is "compatible" with the topology of the space, and most measures used in mathematical analysis and in number theory are indeed Radon measures.
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.
In mathematics, a quasi-invariant measureμ with respect to a transformation T, from a measure space X to itself, is a measure which, roughly speaking, is multiplied by a numerical function of T. An important class of examples occurs when X is a smooth manifold M, T is a diffeomorphism of M, and μ is any measure that locally is a measure with base the Lebesgue measure on Euclidean space. Then the effect of T on μ is locally expressible as multiplication by the Jacobian determinant of the derivative (pushforward) of T.
In mathematics, a regular measure on a topological space is a measure for which every measurable set can be approximated from above by open measurable sets and from below by compact measurable sets.
In mathematics, a positive measure μ defined on a σ-algebra Σ of subsets of a set X is called a finite measure if μ(X) is a finite real number, and a set A in Σ is of finite measure if μ(A) < ∞. The measure μ is called σ-finite if X is the countable union of measurable sets with finite measure. A set in a measure space is said to have σ-finite measure if it is a countable union of measurable sets with finite measure. A measure being σ-finite is a weaker condition than being finite, i.e. all finite measures are σ-finite but there are (many) σ-finite measures that are not finite.
In mathematics, cylinder set measure is a kind of prototype for a measure on an infinite-dimensional vector space. An example is the Gaussian cylinder set measure on Hilbert space.
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.
In mathematics, strict positivity is a concept in measure theory. Intuitively, a strictly positive measure is one that is "nowhere zero", or that is zero "only on points".
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. Alternatively, one may consider Lebesgue measure on finite-dimensional subspaces of the larger space and consider so-called prevalent and shy sets.
In mathematics, an inner regular measure is one for which the measure of a set can be approximated from within by compact subsets.
In mathematics, an invariant measure is a measure that is preserved by some function. Ergodic theory is the study of invariant measures in dynamical systems. The Krylov–Bogolyubov theorem proves the existence of invariant measures under certain conditions on the function and space under consideration.
In measure theory, a discipline within mathematics, a pushforward measure is obtained by transferring a measure from one measurable space to another using a measurable function.