Definition:Space of Measurable Functions
Definition
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} }$
Space of Positive Measurable Functions
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} }$
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.