# Integral of a simple function (measure theory)

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## Definition

For a simple function in its standard representation, say [ilmath]f:=\sum^n_{i=0}x_i\mathbf{1}_{A_i}[/ilmath] then the [ilmath]\mu[/ilmath]-integral, [ilmath]I_\mu:\mathcal{E}^+\rightarrow\mathbb{R} [/ilmath] is[1]:

• $I_\mu(f):=\sum^n_{i=1}x_i\mu(A_i)\in[0,\infty]$

Note that this is independent of the particular standard representation of [ilmath]f[/ilmath].

## Proof of claims

Claim 1: the [ilmath]\mu[/ilmath]-integral of a simple function is independent of which standard representation it is evaluated as.