Definition:Space of Measurable Functions

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
 * Space of Measurable Functions Identified by A.E. Equality