Definition:Measurable Function

Definition
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.