Simple function (measure theory)

From Maths

Jump to: navigation, search

(Unknown grade)

This page is a stub

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:

Needs fleshing out

TODO: Cross reference with Halmos' book

Contents

1 Definition
2 Standard representation
3 References

Definition

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

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

Standard representation

Standard representation (measure theory)/Definition

References

↑ 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, [ilmath]\sigma[/ilmath]-ring, [ilmath]\sigma[/ilmath]-algebra, hereditary [ilmath]\sigma[/ilmath]-ring measurable space

Measures	Pre-measure [ilmath]\leadsto[/ilmath] outer-measure [ilmath]\leadsto[/ilmath] measure (see Extending pre-measures to measures (via: Extending pre-measures to outer-measures [ilmath]\rightarrow[/ilmath] "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 [ilmath]\leadsto[/ilmath] Integral of a positive function [ilmath]\leadsto[/ilmath] Integrating all measurable functions

Retrieved from "https://wiki.unifiedmathematics.com/index.php?title=Simple_function_(measure_theory)&oldid=1908"

Hidden categories:

Navigation menu