Definition:Measurable Function/Extended Real-Valued Function/Definition 2

Definition

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 $\map \BB {\overline \R}$ be the Borel $\sigma$-algebra on the extended real number space.

Let $f : E \to \overline \R$ be an extended real-valued function.

We say that $f$ is ($\Sigma$-)measurable if and only if:

$f$ is $\Sigma_E \, / \, \map \BB {\overline \R}$-measurable.