Editing Measurable function (section)

{{Short description|Kind of mathematical function}}
{{Use American English|date = January 2019}}

In [[mathematics]], and in particular [[Mathematical analysis#Measure_theory|measure theory]], a '''measurable function''' is a function between the underlying sets of two [[measurable space|measurable spaces]] that preserves the structure of the spaces: the [[preimage]] of any [[Measure (mathematics)|measurable]] set is measurable. This is in direct analogy to the definition that a [[Continuous function|continuous]] function between [[topological space|topological spaces]] [[Morphism|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 integration|Lebesgue integral]]. In [[probability theory]], a measurable function on a [[probability space]] is known as a [[random variable]].