Definition:Space of Simple Functions

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

Then the space of simple functions on $\struct {X, \Sigma}$, denoted $\map \EE \Sigma$, is the collection of all simple functions $f: X \to \R$:


 * $\map \EE \Sigma := \set {f: X \to \R: \text{$f$ is a simple function} }$

Space of Positive Simple Functions
The space of positive simple functions on $\struct {X, \Sigma}$, denoted $\map {\EE^+} \Sigma$, is the subset of positive simple functions in $\map \EE \Sigma$:


 * $\map {\EE^+} \Sigma := \set {f: X \to \R: \text {$f$ is a positive simple function} }$

Also known as
Often, one simply speaks about the space of (positive) simple functions when the measurable space $\struct {X, \Sigma}$ is understood.

It is also common to write $\EE$ (resp. $\EE^+$) instead of $\map \EE \Sigma$ (resp. $\map {\EE^+} \Sigma$) when $\Sigma$ is clear from the context.

Also see

 * Simple Function
 * Space of Simple Functions forms Algebra