Definition:Measurable Function

Given a measurable space $$(X, \mathfrak A)\ $$ and a set $$A\in\mathfrak A$$, a function $$f:A\to\R$$ is said to be $$\mathfrak A$$-measurable on $$A\ $$ if $$\{x\in A : f(x) \leq \alpha\}\in\mathfrak A$$ for each $$\alpha\in\R$$.

See the theorem on measurable images for equivalences of this definition.