Lebesgue measure

Definition

The set function [math]\lambda^n:(\mathbb{R}^n,\mathcal{B}(\mathbb{R}^n))\rightarrow\overline{\mathbb{R}_{\ge 0} }[/math] 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]