Difference between revisions of "Lebesgue measure"
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==Definition== | ==Definition== | ||
| − | The set function <math>\lambda^n:(\mathbb{R}^n,\mathcal{B}(\mathbb{R}^n))\rightarrow\mathbb{R}_{\ge}</math><ref>P27 - Measures, Integrals and Martingales - Rene L. Schilling</ref> that assigns every half-open rectangle <math>[[a,b))=[a_1,b_1)\times\cdots\times[a_n,b_n)\in\mathcal{J}</math> as follows: | + | The set function <math>\lambda^n:(\mathbb{R}^n,\mathcal{B}(\mathbb{R}^n))\rightarrow\overline{\mathbb{R}_{\ge 0} }</math><ref>P27 - Measures, Integrals and Martingales - Rene L. Schilling</ref> that assigns every half-open rectangle <math>[\![a,b)\!)=[a_1,b_1)\times\cdots\times[a_n,b_n)\in\mathcal{J}</math> as follows: |
| − | <math>\lambda^n\ | + | <math>\lambda^n\big([\![a,b)\!)\big)=\prod^n_{i=1}(b_i-a_i)</math> |
Where <math>\mathcal{J}=</math> the set of all half-open-half-closed 'rectangles' in <math>\mathbb{R}^n</math> | Where <math>\mathcal{J}=</math> the set of all half-open-half-closed 'rectangles' in <math>\mathbb{R}^n</math> | ||
| − | Note that it can be shown <math>\mathcal{B}(\mathbb{R}^n)=\sigma(\mathcal{J})</math> where <math>\sigma(\mathcal{J})</math> is the [[Sigma-algebra|{{Sigma|algebra}}]] [[Sigma-algebra generated by|generated by]] <math>\mathcal{J}</math> | + | Note that it can be shown <math>\mathcal{B}(\mathbb{R}^n)=\sigma(\mathcal{J})</math> where <math>\sigma(\mathcal{J})</math> is the ''[[Sigma-algebra|{{Sigma|algebra}}]]'' [[Sigma-algebra generated by|generated by]] <math>\mathcal{J}</math> |
{{Definition|Measure Theory}} | {{Definition|Measure Theory}} | ||
==References== | ==References== | ||
Latest revision as of 00:28, 20 December 2016
Definition
The set function [math]\lambda^n:(\mathbb{R}^n,\mathcal{B}(\mathbb{R}^n))\rightarrow\overline{\mathbb{R}_{\ge 0} }[/math][1] that assigns every half-open rectangle [math][\![a,b)\!)=[a_1,b_1)\times\cdots\times[a_n,b_n)\in\mathcal{J}[/math] as follows:
[math]\lambda^n\big([\![a,b)\!)\big)=\prod^n_{i=1}(b_i-a_i)[/math]
Where [math]\mathcal{J}=[/math] the set of all half-open-half-closed 'rectangles' in [math]\mathbb{R}^n[/math]
Note that it can be shown [math]\mathcal{B}(\mathbb{R}^n)=\sigma(\mathcal{J})[/math] where [math]\sigma(\mathcal{J})[/math] is the [ilmath]\sigma[/ilmath]-algebra generated by [math]\mathcal{J}[/math]
References
- ↑ P27 - Measures, Integrals and Martingales - Rene L. Schilling
