In mathematics, more precisely in measure theory, the Lebesgue decomposition theorem [1] provides a way to decompose a measure into two distinct parts based on their relationship with another measure.
The theorem states that if is a measurable space and and are σ-finite signed measures on , then there exist two uniquely determined σ-finite signed measures and such that: [2] [3]
Lebesgue's decomposition theorem can be refined in a number of ways. First, as the Lebesgue-Radon-Nikodym theorem. That is, let be a measure space, a σ-finite positive measure on and a complex measure on . [4]
The first assertion follows from the Lebesgue decomposition, the second is known as the Radon-Nikodym theorem. That is, the function is a Radon-Nikodym derivative that can be expressed as
An alternative refinement is that of the decomposition of a regular Borel measure on the real line [5] [6] where
The absolutely continuous measures are classified by the Radon–Nikodym theorem, and discrete measures are easily understood. Hence (singular continuous measures aside), Lebesgue decomposition gives a very explicit description of measures. The Cantor measure (the probability measure on the real line whose cumulative distribution function is the Cantor function) is an example of a singular continuous measure.
The analogous[ citation needed ] decomposition for a stochastic processes is the Lévy–Itō decomposition: given a Lévy process X, it can be decomposed as a sum of three independent Lévy processes where:
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations of measure are widely used in quantum physics and physics in general.
In linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized. This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces. In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical perspective.
In calculus and real analysis, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central operations of calculus—differentiation and integration. This relationship is commonly characterized in the framework of Riemann integration, but with absolute continuity it may be formulated in terms of Lebesgue integration. For real-valued functions on the real line, two interrelated notions appear: absolute continuity of functions and absolute continuity of measures. These two notions are generalized in different directions. The usual derivative of a function is related to the Radon–Nikodym derivative, or density, of a measure. We have the following chains of inclusions for functions over a compact subset of the real line:
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, the ba space of an algebra of sets is the Banach space consisting of all bounded and finitely additive signed measures on . The norm is defined as the variation, that is
The spectrum of a linear operator that operates on a Banach space is a fundamental concept of functional analysis. The spectrum consists of all scalars such that the operator does not have a bounded inverse on . The spectrum has a standard decomposition into three parts:
In mathematics, a signed measure is a generalization of the concept of (positive) measure by allowing the set function to take negative values, i.e., to acquire sign.
In mathematics, specifically measure theory, a complex measure generalizes the concept of measure by letting it have complex values. In other words, one allows for sets whose size is a complex number.
In measure theory, given a measurable space and a signed measure on it, a set is called a positive set for if every -measurable subset of has nonnegative measure; that is, for every that satisfies holds.
In mathematics, the Bochner integral, named for Salomon Bochner, extends the definition of Lebesgue integral to functions that take values in a Banach space, as the limit of integrals of simple functions.
In mathematics, two positive measures and defined on a measurable space are called singular if there exist two disjoint measurable sets whose union is such that is zero on all measurable subsets of while is zero on all measurable subsets of This is denoted by
In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space , 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 is obtained by summing a large number of independent random variables with variance 1, then has variance and its law is approximately Gaussian.
In mathematics, the disintegration theorem is a result in measure theory and probability theory. It rigorously defines the idea of a non-trivial "restriction" of a measure to a measure zero subset of the measure space in question. It is related to the existence of conditional probability measures. In a sense, "disintegration" is the opposite process to the construction of a product measure.
In mathematics, more precisely in measure theory, a measure on the real line is called a discrete measure if it is concentrated on an at most countable set. The support need not be a discrete set. Geometrically, a discrete measure is a collection of point masses.
In measure theory, a branch of mathematics that studies generalized notions of volumes, an s-finite measure is a special type of measure. An s-finite measure is more general than a finite measure, but allows one to generalize certain proofs for finite measures.
In probability theory and statistics, the law of the unconscious statistician, or LOTUS, is a theorem which expresses the expected value of a function g(X) of a random variable X in terms of g and the probability distribution of X.
In the mathematical theory of random matrices, the Marchenko–Pastur distribution, or Marchenko–Pastur law, describes the asymptotic behavior of singular values of large rectangular random matrices. The theorem is named after soviet mathematicians Volodymyr Marchenko and Leonid Pastur who proved this result in 1967.
In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line which consists of the real numbers and
In mathematics, lifting theory was first introduced by John von Neumann in a pioneering paper from 1931, in which he answered a question raised by Alfréd Haar. The theory was further developed by Dorothy Maharam (1958) and by Alexandra Ionescu Tulcea and Cassius Ionescu Tulcea (1961). Lifting theory was motivated to a large extent by its striking applications. Its development up to 1969 was described in a monograph of the Ionescu Tulceas. Lifting theory continued to develop since then, yielding new results and applications.
In mathematics, the Freudenthal spectral theorem is a result in Riesz space theory proved by Hans Freudenthal in 1936. It roughly states that any element dominated by a positive element in a Riesz space with the principal projection property can in a sense be approximated uniformly by simple functions.
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