Definition:Integral of Positive Simple Function

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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:

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


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