Lebesgue measure

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

where a measurable rectangle is typically a half-open box $R=\prod_{i=1}^n (a_i,b_i]$ and

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 $0$.