For questions about measurable functions.
A measurable function is a structure preserving map between measurable spaces. This means that the inverse image of a measurable subset of the codomain is a measurable subset of the domain. It 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.
