Definition:Space of Measurable Functions/Positive

Definition

Space of Positive Real-Valued Measurable Functions

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

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

That is:

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

Space of Positive Extended Real-Valued Measurable Functions

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} }$