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Monotone class theorem information


In measure theory and probability, the monotone class theorem connects monotone classes and 𝜎-algebras. The theorem says that the smallest monotone class containing an algebra of sets is precisely the smallest 𝜎-algebra containing  It is used as a type of transfinite induction to prove many other theorems, such as Fubini's theorem.

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Monotone class theorem

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probability, the monotone class theorem connects monotone classes and 𝜎-algebras. The theorem says that the smallest monotone class containing an algebra...

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Monotone

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rules and bargaining systems. Monotone class theorem, in measure theory Monotone convergence theorem, in mathematics Monotone polygon, a property of a geometric...

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List of theorems

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Mohr–Mascheroni theorem (geometry) Monge's theorem (geometry) Monodromy theorem (complex analysis) Monotone class theorem (measure theory) Monotone convergence...

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Monotone likelihood ratio

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D.; Cohen, Arthur; Strawderman, W.E. (1976). "A complete class theorem for strict monotone likelihood ratio with applications". Annals of Statistics...

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Absolutely and completely monotonic functions and sequences

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Threshold Logic". IEEE Transactions on Computers. 17 (3). Bernstein's theorem on monotone functions Hausdorff moment problem Monotonic function Cyclical monotonicity...

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Monotone comparative statics

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Monotone comparative statics is a sub-field of comparative statics that focuses on the conditions under which endogenous variables undergo monotone changes...

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Hereditary property

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result from this area is the unique factorization theorem. There is no consensus for the meaning of "monotone property" in graph theory. Examples of definitions...

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Point process

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^{n}(A_{1}\times \cdots \times A_{n})=\prod _{i=1}^{n}\xi (A_{i})} By monotone class theorem, this uniquely defines the product measure on ( S n , B ( S n )...

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Dynkin system

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concept in measure theory, also referred to as an algebra of sets Monotone class – theoremPages displaying wikidata descriptions as a fallbackPages displaying...

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Envelope theorem

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In mathematics and economics, the envelope theorem is a major result about the differentiability properties of the value function of a parameterized optimization...

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Ring of sets

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Family closed under complements and countable disjoint unions Monotone class – theoremPages displaying wikidata descriptions as a fallbackPages displaying...

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Trace inequality

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and operator monotone, while f ( t ) = t log ⁡ ( t ) {\displaystyle f(t)=t\log(t)} is operator convex. The original proof of this theorem is due to K....

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Galois connection

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depending on whether we take monotone or antitone Galois connections). Let (A, ≤) and (B, ≤) be two partially ordered sets. A monotone Galois connection between...

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Art gallery problem

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shown to be true for vertex guards in two special sub-classes of simple polygons, viz. monotone polygons and polygons weakly visible from an edge. Krohn...

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Contraction mapping

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non-empty closed convex sets. The class of firmly nonexpansive operators is equal to the set of resolvents of maximally monotone operators. Surprisingly, while...

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Field of sets

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volumePages displaying short descriptions of redirect targets Monotone class – theoremPages displaying wikidata descriptions as a fallbackPages displaying...

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Descriptive complexity theory

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fixed-point of a monotone expression. This augments first-order logic with the ability to express recursion. The Immerman–Vardi theorem, shown independently...

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Dirichlet function

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Dirichlet function provides a counterexample showing that the monotone convergence theorem is not true in the context of the Riemann integral. Proof Using...

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