\begin{definition}[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.
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\end{definition}