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Decomposable Measure
In mathematics, a decomposable measure (also known as a strictly localizable measure) is a measure that is a disjoint union of finite measures. This is a generalization of σ-finite measures, which are the same as those that are a disjoint union of ''countably many'' finite measures. There are several theorems in measure theory such as the Radon–Nikodym theorem that are not true for arbitrary measures but are true for σ-finite measures. Several such theorems remain true for the more general class of decomposable measures. This extra generality is not used much as most decomposable measures that occur in practice are σ-finite. Examples * Counting measure In mathematics, specifically measure theory, the counting measure is an intuitive way to put a measure on any set – the "size" of a subset is taken to be the number of elements in the subset if the subset has finitely many elements, and infinity ... on an uncountable measure space with all subsets measurable is a decomposabl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measure (mathematics)
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as 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 (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Const ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disjoint Union
In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A (that is, each element of A belongs to exactly one of these images). A disjoint union of a family of pairwise disjoint sets is their union. In category theory, the disjoint union is the coproduct of the category of sets, and thus defined up to a bijection. In this context, the notation \coprod_ A_i is often used. The disjoint union of two sets A and B is written with infix notation as A \sqcup B. Some authors use the alternative notation A \uplus B or A \operatorname B (along with the corresponding \biguplus_ A_i or \operatorname_ A_i). A standard way for building the disjoint union is to define A as the set of ordered pairs (x, i) such that x \in A_i, and the injection A_i \to A as x \mapsto (x, i). Example Consider the sets A_0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Measure
In measure theory, a branch of mathematics, a finite measure or totally finite measure is a special measure that always takes on finite values. Among finite measures are probability measures. The finite measures are often easier to handle than more general measures and show a variety of different properties depending on the sets they are defined on. Definition A measure \mu on measurable space (X, \mathcal A) is called a finite measure iff it satisfies : \mu(X) < \infty. By the monotonicity of measures, this implies : If is a finite measure, the is called a finite measure space or a totally finite measure space. Properties General case For any meas ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
σ-finite Measure
In mathematics, a positive (or signed) 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 (rather than ∞), and a set ''A'' in Σ is of finite measure if ''μ''(''A'') < ∞''.'' The measure ''μ'' is called σ-finite if ''X'' is a 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. A different but related notion that should not be confus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Countably Many
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements. In more technical terms, assuming the axiom of countable choice, a set is ''countable'' if its cardinality (its number of elements) is not greater than that of the natural numbers. A countable set that is not finite is said countably infinite. The concept is attributed to Georg Cantor, who proved the existence of uncountable sets, that is, sets that are not countable; for example the set of the real numbers. A note on terminology Although the terms "countable" and "countably infinite" as defined here are quite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measure Theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as 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 (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Const ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radon–Nikodym Theorem
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. One way to derive a new measure from one already given is to assign a density to each point of the space, then integrate over the measurable subset of interest. This can be expressed as :\nu(A) = \int_A f \, d\mu, where is the new measure being defined for any measurable subset and the function is the density at a given point. The integral is with respect to an existing measure , which may often be the canonical Lebesgue measure on the real line or the ''n''-dimensional Euclidean space (corresponding to our sta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Counting Measure
In mathematics, specifically measure theory, the counting measure is an intuitive way to put a measure on any set – the "size" of a subset is taken to be the number of elements in the subset if the subset has finitely many elements, and infinity \infty if the subset is infinite. The counting measure can be defined on any measurable space (that is, any set X along with a sigma-algebra) but is mostly used on countable sets. In formal notation, we can turn any set X into a measurable space by taking the power set of X as the sigma-algebra \Sigma; that is, all subsets of X are measurable sets. Then the counting measure \mu on this measurable space (X,\Sigma) is the positive measure \Sigma \to ,+\infty/math> defined by \mu(A) = \begin \vert A \vert & \text A \text\\ +\infty & \text A \text \end for all A\in\Sigma, where \vert A\vert denotes the cardinality of the set A. The counting measure on (X,\Sigma) is σ-finite if and only if the space X is countable In mathematics, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fubini's Theorem
In mathematical analysis Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if the double integral yields a finite answer when the integrand is replaced by its absolute value. \, \iint\limits_ f(x,y)\,\text(x,y) = \int_X\left(\int_Y f(x,y)\,\texty\right)\textx=\int_Y\left(\int_X f(x,y) \, \textx \right) \texty \qquad \text \qquad \iint\limits_ , f(x,y), \,\text(x,y) <+\infty. Fubini's theorem implies that two iterated integrals are equal to the corresponding double integral across its integrands. Tonelli's theorem, introduced by in 1909, is similar, but applies to a non-negative measurable function rather than one integrable over their domains. A related theorem is oft ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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