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 f:X→R on a measurable space (X,A) is a[1]:
- function of the form ∑Ni=1xi1Ai(x) for
- finitely many sets, A1,…,AN∈A and
- finitely many x1,…,xn∈R
