Definition:Standard Representation of Simple Function

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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.

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