Simple function (measure theory)

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TODO: Cross reference with Halmos' book

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Definition

A simple function $f:X\rightarrow\mathbb{R}$ on a measurable space $(X,\mathcal{A})$ is a:

• function of the form $\sum^N_{i=1}x_i\mathbf{1}_{A_i}(x)$ for
• finitely many sets, $A_1,\ldots,A_N\in\mathcal{A}$ and
• finitely many $x_1,\ldots,x_n\in\mathbb{R}$

Standard representation

Standard representation (measure theory)/Definition