Measure theory | Types of functions

Measurable function

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. (Wikipedia).

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Related pages

Topological space | Non-measurable set | Indicator function | Probability space | Continuous function | Mathematical analysis | Lebesgue integration | Measurable space | Zermelo–Fraenkel set theory | Borel set | Pointwise | Real analysis | Mathematics | Morphism | Random variable | Complex number | Σ-algebra | Probability theory | Lp space | Measure (mathematics) | Open set