Lebesgue measure - Knowlpedia

A Lebesgue measure on $\mathbb R^n$ is the complete measure $\lambda^n$ obtained from the outer measure

\lambda^{n,*}(E) =\inf\left\{\sum_{k=1}^\infty \operatorname{vol}(R_k)\,:\, E\subseteq \bigcup_{k=1}^\infty R_k,\ \text{each } R_k \text{ is a measurable rectangle}\right\},

where the $R_k$ are half-open boxes and

\operatorname{vol}(R)=\prod_{i=1}^n (b_i-a_i).

A set $E\subseteq\mathbb R^n$ is Lebesgue measurable if it is Carathéodory measurable for $\lambda^{n,*}$ , and then $\lambda^n(E)=\lambda^{n,*}(E)$ .

This construction yields the unique complete, translation-invariant measure on the Lebesgue measurable sets normalized by $\lambda^n([0,1]^n)=1$ .

Examples

On $\mathbb R$ , $\lambda^1((a,b))=b-a$ .
Every countable subset of $\mathbb R^n$ has Lebesgue measure $0$ .