View source for Lebesgue measure



==Definition==
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\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 ''[[Sigma-algebra|{{Sigma|algebra}}]]'' [[Sigma-algebra generated by|generated by]] <math>\mathcal{J}</math>

{{Definition|Measure Theory}}
==References==