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In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in direct analogy to the definition that a continuous function between topological spaces preserves the topological structure: the preimage of any open set is open. In real analysis, measurable functions are used in the definition of the Lebesgue integral. In probability theory, a measurable function on a probability space is known as a random variable.

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rdfs:label	Measurable function (en)
rdfs:comment	In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in direct analogy to the definition that a continuous function between topological spaces preserves the topological structure: the preimage of any open set is open. In real analysis, measurable functions are used in the definition of the Lebesgue integral. In probability theory, a measurable function on a probability space is known as a random variable. (en)
sameAs	Measurable function Measurable function
dbp:wikiPageUsesTemplate	dbt:Annotated_link dbt:Reflist dbt:Short_description dbt:Use_American_English dbt:Measure_theory
Subject	Types of functions Measure theory
prov:wasDerivedFrom	http://en.wikipedia.org/wiki/Measurable_function?oldid=1070275505&ns=0
Wikipage page ID	44055 (xsd:integer)
page length (characters) of wiki page	9005 (xsd:nonNegativeInteger)
Wikipage revision ID	1070275505 (xsd:integer)
Link from a Wikipage to another Wikipage	Non-measurable set Probability space Probability theory Axiom of choice Bochner measurable function Borel set Σ-algebra Types of functions Measure theory Dimension (vector space) Zermelo–Fraenkel set theory Complex number Continuous function Morphism Open set Encyclopedia of Mathematics Image (mathematics) Indicator function Infimum and supremum Weakly measurable function Absolutely integrable function Topological space Lusin's theorem Lebesgue integration Lebesgue measure Limit inferior and limit superior Numerical analysis Mathematics Measurable space Measure (mathematics) Pointwise Random variable Real analysis
has abstract	In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in direct analogy to the definition that a continuous function between topological spaces preserves the topological structure: the preimage of any open set is open. In real analysis, measurable functions are used in the definition of the Lebesgue integral. In probability theory, a measurable function on a probability space is known as a random variable. (en)
foaf:isPrimaryTopicOf	http://en.wikipedia.org/wiki/Measurable_function
is rdfs:seeAlso of	Real-valued function
is Wikipage redirect of	Borel function Borel section Measureable function Measurable map Measurable mapping Lebesgue-measurable function Lebesgue measurable function
is Link from a Wikipage to another Wikipage of	Forcing (mathematics) Grothendieck inequality List of types of functions Locally compact space Locally integrable function Maharam's theorem Norm (mathematics) Absolute convergence Absolutely integrable function Adapted process Alexandra Bellow Alexandru Froda Azuma's inequality Baire set Base flow (random dynamical systems) Besicovitch covering theorem Better-quasi-ordering Blichfeldt's theorem Borel equivalence relation Borel function Borel functional calculus Borel section Borel set Monotone convergence theorem Mixed binomial process Bounded variation Campbell's theorem (probability) Carathéodory's existence theorem Cardinality of the continuum Giovanni Alberti (mathematician) Giuseppe Vitali Diamagnetic inequality Discontinuous linear map Disintegration theorem Dominated convergence theorem Doob decomposition theorem Doob martingale Doob–Dynkin lemma Characterizations of the exponential function Chebyshev's inequality Clarkson's inequalities Cuspidal representation

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