Difference between revisions of "Simple function (measure theory)"

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(Created page with "{{Stub page|Needs fleshing out}} {{Todo|Cross reference with Halmos' book}} __TOC__ ==Definition== A ''simple function'' {{M|f:X\rightarrow\mathbb{R} }} on a measurable spac...")
 
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{{Todo|Cross reference with Halmos' book}}
 
{{Todo|Cross reference with Halmos' book}}
 
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==Definition==
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==[[Simple function (measure theory)/Definition|Definition]]==
A ''simple function'' {{M|f:X\rightarrow\mathbb{R} }} on a [[measurable space]] {{M|(X,\mathcal{A})}} is a{{rMIAMRLS}}:
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{{:Simple function (measure theory)/Definition}}
* function of the form {{M|1=\sum^N_{i=1}x_i\mathbf{1}_{A_i}(x)}} for
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* finitely many sets, {{M|A_1,\ldots,A_N\in\mathcal{A} }} and
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* finitely many {{M|x_1,\ldots,x_n\in\mathbb{R} }}
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==[[Standard representation (measure theory)|Standard representation]]==
 
==[[Standard representation (measure theory)|Standard representation]]==
 
{{:Standard representation (measure theory)/Definition}}
 
{{:Standard representation (measure theory)/Definition}}

Latest revision as of 07:16, 12 March 2016

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Needs fleshing out

TODO: Cross reference with Halmos' book


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

  1. Measures, Integrals and Martingales - René L. Schilling