Definition:Space of Simple Functions

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

Then the space of simple functions on $\left({X, \Sigma}\right)$, denoted $\mathcal E \left({\Sigma}\right)$, is the collection of all simple functions $f: X \to \R$:


 * $\mathcal E \left({\Sigma}\right) := \left\{{f: X \to \R: f \text{ is a simple function}}\right\}$

Space of Positive Simple Functions
The space of positive simple functions on $\left({X, \Sigma}\right)$, denoted $\mathcal{E}^+ \left({\Sigma}\right)$ is the subset of positive simple functions in $\mathcal E \left({\Sigma}\right)$:


 * $\mathcal{E}^+ \left({\Sigma}\right) := \left\{{f: X \to \R: f \text{ is a positive simple function}}\right\}$

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

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

Also see

 * Simple Function
 * Space of Simple Functions forms Ring