Simple function (measure theory)

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TODO: Cross reference with Halmos' book

Contents

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1 Definition
2 Standard representation
3 References

Definition

A simple function $f:X\rightarrow\mathbb{R}$ on a measurable space $(X,\mathcal{A})$ is a^[1]:

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

References

Jump up ↑ Measures, Integrals and Martingales - René L. Schilling

v • d • e Measure Theory

Overview of the basic and important objects studied in measure theory

Set-structures	ring of sets, algebra of sets, $\sigma$ -ring, $\sigma$ -algebra, hereditary $\sigma$ -ring measurable space

Measures	Pre-measure $\leadsto$ outer-measure $\leadsto$ measure (see Extending pre-measures to measures (via: Extending pre-measures to outer-measures $\rightarrow$ "Restricting outer-measures to measures" maybe?). Alternatively: inner-measure can be used. measure space

Functions	measurable map, simple function (standard representation)

Spaces

Integrals	Integrating a simple function $\leadsto$ Integral of a positive function $\leadsto$ Integrating all measurable functions

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