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:

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

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.

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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)