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Sub-probability Measure
In the mathematical theory of probability and measure, a sub-probability measure is a measure that is closely related to probability measures. While probability measures always assign the value 1 to the underlying set, sub-probability measures assign a value lesser than or equal to 1 to the underlying set. Definition Let \mu be a measure on the measurable space (X, \mathcal A) . Then \mu is called a sub-probability measure if \mu(X) \leq 1 . Properties In measure theory, the following implications hold between measures: \text \implies \text \implies \text \implies \sigma\text So every probability measure is a sub-probability measure, but the converse is not true. Also every sub-probability measure is a finite measure and a σ-finite measure, but the converse is again not true. See also * Helly's selection theorem * Helly–Bray theorem In probability theory, the Helly–Bray theorem relates the weak convergence of cumulative distribution functions t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms of probability, axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure (mathematics), measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event (probability theory), event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of determinism, non-deterministic or uncertain processes or measured Quantity, quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly p ... [...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] |
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, Co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Measure
In 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''. The difference between a probability measure and the more general notion of measure (which includes concepts like area or volume) is that a probability measure must assign value 1 to the entire probability space. Intuitively, the additivity property says that the probability assigned to the union of two disjoint events by the measure should be the sum of the probabilities of the events; for example, the value assigned to "1 or 2" in a throw of a dice should be the sum of the values assigned to "1" and "2". Probability measures have applications in diverse fields, from physics to finance and biology. Definition The requirements for a function \mu to be a probability measure on a probability space are that: * \mu must return results in the unit interval , 1 returning 0 for the empty set and 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measurable Space
In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. Definition Consider a set X and a σ-algebra \mathcal A on X. Then the tuple (X, \mathcal A) is called a measurable space. Note that in contrast to a measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that i ..., no measure is needed for a measurable space. Example Look at the set: X = \. One possible \sigma-algebra would be: \mathcal A_1 = \. Then \left(X, \mathcal A_1\right) is a measurable space. Another possible \sigma-algebra would be the power set on X: \mathcal A_2 = \mathcal P(X). With this, a second measurable space on the set X is given by \left(X, \mathcal A_2\right). Common measurable ... [...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 measure space is called a finite measure space or a totally finite measure space. Properties General case For any measurable space, the finite ...[...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 c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helly's Selection Theorem
In mathematics, Helly's selection theorem (also called the ''Helly selection principle'') states that a uniformly bounded sequence of monotone real functions admits a Convergent sequence, convergent subsequence. In other words, it is a sequential compactness theorem for the space of uniformly bounded monotone functions. It is named for the Austrian mathematician Eduard Helly. A more general version of the theorem asserts compactness of the space BVloc of functions locally of Bounded variation, bounded total variation that are uniformly bounded at a point. The theorem has applications throughout mathematical analysis. In probability theory, the result implies compactness of a tightness of measures, tight family of measures. Statement of the theorem Let (''f''''n'')''n'' ∈ N be a sequence of increasing functions mapping the real line R into itself, and suppose that it is uniformly bounded: there are ''a,b'' ∈ R such that ''a'' ≤ ''f''''n'' � ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helly–Bray Theorem
In probability theory, the Helly–Bray theorem relates the weak convergence of cumulative distribution functions to the convergence of expectations of certain measurable functions. It is named after Eduard Helly and Hubert Evelyn Bray. Let ''F'' and ''F''1, ''F''2, ... be cumulative distribution functions on the real line. The Helly–Bray theorem states that if ''F''''n'' converges weakly to ''F'', then ::\int_\mathbb g(x)\,dF_n(x) \quad\xrightarrow \to\inftyquad \int_\mathbb g(x)\,dF(x) for each bounded, continuous function ''g'': R → R, where the integrals involved are Riemann–Stieltjes integrals. Note that if ''X'' and ''X''1, ''X''2, ... are random variables corresponding to these distribution functions, then the Helly–Bray theorem does not imply that E(''X''''n'') → E(''X''), since ''g''(''x'') = ''x'' is not a bounded function. In fact, a stronger and more general theorem holds. Let ''P'' and ''P''1, ''P''2, ... be probability measure In mathem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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