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

Definition

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

Then the space of $\Sigma$-measurable, positive extended real-valued functions $\map {\MM_{\overline \R}^+} \Sigma$ is the subset of $\map {\MM^+} \Sigma$ consisting of the positive $\Sigma$-measurable functions in $\map {\MM_{\overline \R} } \Sigma$.

That is:

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

Also see

• Space of Extended Real-Valued Measurable Functions