For a subset $A$ of the \reftext{def:real-numbers-c54-2026c}{real line} $\mathbb{R}$, the \textbf{Lebesgue outer measure} of $A$ is

where the \reftext{def:lower-bound-infimum-c54-2026a}{infimum} is taken over all \reftext{def:sequence-in-set-2026a}{sequences} of open \reftext{def:interval-real-line-c54-2026c}{intervals} $(a_m,b_m)$ with $a_m\le b_m$ whose union contains $A$, the sum is understood as in \ref{def:measure-measure-space-2026a}, and $\lambda^{*}(A)=\infty$ if no such sequence yields a finite sum. Since $\mathbb{R}$ is covered by the intervals $(-m,m)$, at least one covering sequence always exists, so $\lambda^{*}(A)\in[0,\infty]$ is defined for every $A\subseteq\mathbb{R}$.

That $\lambda^{*}$ is an \reftext{def:outer-measure-2026a}{outer measure} is the content of claim 1 of \ref{thm:lebesgue-measure-real-line-2026a}.