Definition:Space of Measurable Functions

Definition
Let $\left({X, \Sigma}\right)$ be a measurable space.

Then the space of $\Sigma$-measurable, real-valued functions $\mathcal M \left({\Sigma}\right)$ is the collection of all $\Sigma$-measurable, real-valued functions:


 * $\mathcal M \left({\Sigma}\right) := \left\{{f: X \to \R: f \text{ is $\Sigma$-measurable}}\right\}$

Similarly, the space of $\Sigma$-measurable, extended real-valued functions $\mathcal{M}_{\overline{\R}} \left({\Sigma}\right)$ is the collection of all $\Sigma$-measurable, extended real-valued functions:


 * $\mathcal{M}_{\overline{\R}} \left({\Sigma}\right) := \left\{{f: X \to \overline{\R}: f \text{ is $\Sigma$-measurable}}\right\}$

Space of Positive Measurable Functions
The space of $\Sigma$-measurable, positive real-valued functions $\mathcal{M}^+ \left({\Sigma}\right)$ is the subset of positive $\Sigma$-measurable functions in $\mathcal M \left({\Sigma}\right)$:


 * $\mathcal{M}^+ \left({\Sigma}\right) := \left\{{f: X \to \R: f \text{ is positive $\Sigma$-measurable}}\right\}$

Analogously, the space of $\Sigma$-measurable, positive extended real-valued functions $\mathcal{M}_{\overline{\R}}^+ \left({\Sigma}\right)$ is defined as:


 * $\mathcal{M}_{\overline{\R}}^+ \left({\Sigma}\right) := \left\{{f: X \to \overline{\R}: f \text{ is positive $\Sigma$-measurable}}\right\}$

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 $\mathcal M \left({\Sigma}\right)$ or $\mathcal{M}_{\overline{\R}} \left({\Sigma}\right)$, 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 $\mathcal M$ or $\mathcal{M}_{\overline{\R}}$ 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