Definition:Integral of Positive Simple Function

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

Let $\mathcal E^+$ denote the space of positive simple functions

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

Suppose that $f$ admits the following standard representation:


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

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

Then the $\mu$-integral of $f$, denoted $I_\mu \left({f}\right)$, is defined by:


 * $I_\mu \left({f}\right) := \displaystyle \sum_{i \mathop = 0}^n a_i \mu \left({E_i}\right)$

Also see

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