Definition:Standard Representation of Simple Function

Definition

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

Let $f: X \to \R$ be a simple function.

A standard representation of $f$ consists of:

a finite sequence $a_0, \ldots, a_n$ of real numbers

a partition $E_0, E_1, \ldots, E_n$ of $\Sigma$-measurable sets

subject to:

$f = \ds \sum_{j \mathop = 0}^n a_j \chi_{E_j}$

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

Also see

• Simple Function has Standard Representation

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Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $8.6$