\begin{definition}[Definition:Measurable Function/Real-Valued Function/Definition 2]
Let $\struct {X, \Sigma}$ be a measurable space.
Let $E \in \Sigma$.
Let $\Sigma_E$ be the trace $\sigma$-algebra of $E$ in $\Sigma$. 
Let $\map \BB \R$ be the Borel $\sigma$-algebra on $\R$.
Let $f : E \to \R$ be a real-valued function.
We say that $f$ is '''($\Sigma$-)measurable''' {{iff}}: 
:$f$ is $\Sigma_E \, / \, \map \BB \R$-measurable.
Category:Definitions/Measurable Functions
\end{definition}