Category:Definitions/Space of Measurable Functions

This category contains definitions related to Space of Measurable Functions.
Related results can be found in Category:Space of Measurable Functions.

Space of Real-Valued Measurable Functions

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

Then the space of $\Sigma$-measurable, real-valued functions $\map \MM {X, \Sigma, \R}$ is the set of all $\Sigma$-measurable, real-valued functions.

That is:

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

Space of Extended Real-Valued Measurable Functions

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

Then the space of $\Sigma$-measurable, extended real-valued functions $\map {\MM_{\overline \R}} {X, \Sigma}$ is the set of all $\Sigma$-measurable, extended real-valued functions.

That is:

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