Let $\lambda^{*}$ be the \reftext{def:lebesgue-outer-measure-real-line-2026a}{Lebesgue outer measure} on the real line. Then:

1. $\lambda^{*}$ is an \reftext{def:outer-measure-2026a}{outer measure} on $\mathbb{R}$;
2. every \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel set} is Carathéodory measurable with respect to $\lambda^{*}$;
3. consequently, by \ref{thm:caratheodory-extension-2026a}, the restriction $\lambda$ of $\lambda^{*}$ to $\mathcal{B}(\mathbb{R})$ is a \reftext{def:measure-measure-space-2026a}{measure}, called \textbf{Lebesgue measure} on $\mathbb{R}$;
4. for all real $a\le b$,

and more generally every \reftext{def:interval-real-line-c54-2026c}{interval} with endpoints $a\le b$ has Lebesgue measure $b-a$; 5. $\lambda$ is $\sigma$-finite.