Definition:Space of Measurable Functions/Extended Real-Valued

Definition

Let $\struct {X, \Sigma}$ be a measurable space.

Then the space of $\Sigma$-measurable, extended real-valued functions $\map {\MM_{\overline \R}} {X, \Sigma}$ is the set of all $\Sigma$-measurable, extended real-valued functions.

That is:

$\map {\MM_{\overline \R}} {X, \Sigma} := \set {f: X \to \overline \R: f \text{ is $\Sigma$-measurable} }$

Also see

• Definition:Space of Positive Extended Real-Valued Measurable Functions

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $8.4$