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

Sources

• 1991: David Williams: Probability with Martingales ... (previous) ... (next): $5.1$: Integrals of non-negative simple functions, $SF^+$
• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $9.2$