Difference between revisions of "Simple function (measure theory)/Definition"
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(Created page with "<noinclude> ==Definition== </noinclude> A ''simple function'' {{M|f:X\rightarrow\mathbb{R} }} on a measurable space {{M|(X,\mathcal{A})}} is a{{rMIAMRLS}}: * function of t...") |
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| − | </noinclude> | + | </noinclude>A ''simple function'' {{M|f:X\rightarrow\mathbb{R} }} on a [[measurable space]] {{M|(X,\mathcal{A})}} is a{{rMIAMRLS}}: |
| − | A ''simple function'' {{M|f:X\rightarrow\mathbb{R} }} on a [[measurable space]] {{M|(X,\mathcal{A})}} is a{{rMIAMRLS}}: | + | |
* function of the form {{M|1=\sum^N_{i=1}x_i\mathbf{1}_{A_i}(x)}} for | * function of the form {{M|1=\sum^N_{i=1}x_i\mathbf{1}_{A_i}(x)}} for | ||
* finitely many sets, {{M|A_1,\ldots,A_N\in\mathcal{A} }} and | * finitely many sets, {{M|A_1,\ldots,A_N\in\mathcal{A} }} and | ||
Latest revision as of 17:06, 17 March 2016
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]
