Definition:Space of Measurable Functions

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

Then the space of $\Sigma$-measurable, real-valued functions $\map \MM \Sigma$ is the collection of all $\Sigma$-measurable, real-valued functions:


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

Similarly, the space of $\Sigma$-measurable, extended real-valued functions $\map {\MM_{\overline \R} } \Sigma$ is the collection of all $\Sigma$-measurable, extended real-valued functions:


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

Space of Positive Measurable Functions
The space of $\Sigma$-measurable, positive real-valued functions $\map {\MM^+} \Sigma$ is the subset of positive $\Sigma$-measurable functions in $\map \MM \Sigma$:


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

Analogously, the space of $\Sigma$-measurable, positive extended real-valued functions $\map {\MM_{\overline \R}^+} \Sigma$ is defined as:


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

Also known as
It is often taken understood from the notation whether the functions are real-valued or extended real-valued.

Thus, one often speaks about the space of $\Sigma$-measurable functions, which can mean either $\map \MM \Sigma$ or $\map {\MM_{\overline \R} } \Sigma$, depending on the context.

When the $\sigma$-algebra $\Sigma$ is clear from the context, it may be dropped both from name and notation.

For example, one would write simply $\MM$ or $\MM_{\overline \R}$ and call it the space of measurable functions.

For any of these notations, adding a superscript $+$ indicates the space of positive measurable functions.

Also see

 * Measurable Function