Difference between revisions of "Lebesgue measure"

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(Created page with " ==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 - Re...")
 
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<math>\lambda^n\Big([[a,b))\Big)=\Pi^n_{i=1}(b_i-a_i)</math>
 
<math>\lambda^n\Big([[a,b))\Big)=\Pi^n_{i=1}(b_i-a_i)</math>
  
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Where <math>\mathcal{J}=</math> the set of all half-open-half-closed 'rectangles' in <math>\mathbb{R}^n</math>
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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==

Revision as of 18:32, 15 March 2015


Definition

The set function [math]\lambda^n:(\mathbb{R}^n,\mathcal{B}(\mathbb{R}^n))\rightarrow\mathbb{R}_{\ge}[/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)=\Pi^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 [[Sigma-algebra|[ilmath]\sigma[/ilmath]-algebra generated by [math]\mathcal{J}[/math]

References

  1. P27 - Measures, Integrals and Martingales - Rene L. Schilling