Definition:Measurable Function/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 $\map \BB \R$ be the Borel $\sigma$-algebra on $\R$.

Let $f : E \to \R$ be a real-valued function.

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

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

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 8$