Definition:Measurable Function

Let $$\left({X, \mathfrak A}\right)\ $$ be a measurable space.

Let $$A \in \mathfrak A$$.

Then a function $$f: A \to \R$$ is said to be $$\mathfrak A$$-measurable on $$A\ $$ if:


 * $$\forall \alpha \in \R: \left\{{x \in A : f \left({x}\right) \le \alpha}\right\} \in \mathfrak A$$

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