Integral of a simple function (measure theory)

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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 $f$.

Proof of claims

[Expand]

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

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The integral of a simple function is independent of the standard representation used to evaluate it/Proof

See next

• Integral of a positive function (measure theory)

See also

• Integral (measure theory)