Definition:Integral of Positive Simple Function

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

Let $\EE^+$ denote the space of positive simple functions.

Let $f: X \to \R, f \in \EE^+$ be a positive simple function.

Suppose that $f$ admits the following standard representation:


 * $\ds f = \sum_{i \mathop = 0}^n a_i \chi_{E_i}$

where $a_0 := 0$, and $\chi$ denotes characteristic function.

Then the $\mu$-integral of $f$, denoted $\map {I_\mu} f$, is defined by:


 * $\ds \map {I_\mu} f := \sum_{i \mathop = 0}^n a_i \map \mu {E_i}$

Also see

 * Integral of Positive Simple Function is Well-Defined, ensuring well-definition of $\map {I_\mu} f$
 * Simple Function has Standard Representation, ensuring $\map {I_\mu} f$ is always defined