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Statements

Subject Item

dbr:Measurable_function

rdfs:label

Measurable function

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.

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dbc:Types_of_functions dbc:Measure_theory

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n9:Measurable_function?oldid=1070275505&ns=0

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dbr:Morphism dbr:Mathematical_analysis dbr:Measure_(mathematics) dbc:Types_of_functions dbr:Measurable_space dbr:Indicator_function dbr:Lp_space dbr:Lebesgue_integration dbr:Lebesgue_measure dbr:Random_variable dbr:Borel_set dbr:Pointwise dbr:Dimension_(vector_space) dbr:Probability_theory dbr:Open_set dbr:Image_(mathematics) dbr:Probability_space dbr:Topological_space dbr:Bochner_measurable_function dbr:Weakly_measurable_function dbr:Lusin's_theorem dbr:Zermelo–Fraenkel_set_theory dbr:Mathematics dbr:Σ-algebra dbr:Encyclopedia_of_Mathematics dbr:Limit_inferior_and_limit_superior dbr:Continuous_function dbr:Axiom_of_choice dbr:Real_analysis dbr:Non-measurable_set dbc:Measure_theory dbr:Complex_number dbr:Infimum_and_supremum

dbo: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.

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n9:Measurable_function