Lebesgue measure

Definition

The set function

$$\lambda^n:(\mathbb{R}^n,\mathcal{B}(\mathbb{R}^n))\rightarrow\overline{\mathbb{R}_{\ge 0} }$$ that assigns every half-open rectangle $$[\![a,b)\!)=[a_1,b_1)\times\cdots\times[a_n,b_n)\in\mathcal{J}$$ as follows:

$$\lambda^n\big([\![a,b)\!)\big)=\prod^n_{i=1}(b_i-a_i)$$

Where

$$\mathcal{J}=$$ the set of all half-open-half-closed 'rectangles' in $$\mathbb{R}^n$$

Note that it can be shown

$$\mathcal{B}(\mathbb{R}^n)=\sigma(\mathcal{J})$$ where $$\sigma(\mathcal{J})$$ is the $\sigma$-algebra generated by $$\mathcal{J}$$