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.

Non-measurable functions

Real-valued functions encountered in applications tend to be measurable;
however, it is not difficult to prove the existence of non-measurable functions.

Such proofs rely on the axiom of choice in an essential way, in the sense that
Zermeloโ€“Fraenkel set theory without the axiom of choice does not prove
the existence of such functions.