Integral of a simple function (measure theory)/Definition

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Quickly written. Needs to go something like "using every simple function has a standard representation" and then showing that the integral is the same for each standard representation is what needs to be done

Definition

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

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

Note that this is independent of the particular standard representation of $f$.