Lebesgue measure - Imported Knowlpedia Content

A Lebesgue measure on $\mathbb{R}^n$ is the measure $\lambda^n$ obtained by applying the Carathéodory construction to the outer measure $\lambda^{n,*}$ defined by

\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 a measurable rectangle is typically a half-open box $R=\prod_{i=1}^n (a_i,b_i]$ 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 a Carathéodory measurable set for $\lambda^{n,*}$ , and then $\lambda^n(E):=\lambda^{n,*}(E)$ .

Lebesgue measure is the foundational example of a measure space on Euclidean space and is the standard reference for notions like null set and almost everywhere.

Examples

On $\mathbb{R}$ , $\lambda^1((a,b))=b-a$ for any open interval $(a,b)$ .
In $\mathbb{R}^n$ , $\lambda^n\!\left(\prod_{i=1}^n (a_i,b_i]\right)=\prod_{i=1}^n (b_i-a_i)$ for any measurable rectangle.
Any countable subset of $\mathbb{R}^n$ (for example, $\mathbb{Q}\cap[0,1]$ ) has Lebesgue measure $0$ .