Lebesgue Outer Measure on the Real Line

For a subset $A$ of the real line $\mathbb{R}$, the Lebesgue outer measure of $A$ is

where the infimum is taken over all sequences of open intervals $(a_m,b_m)$ with $a_m\le b_m$ whose union contains $A$, the sum is understood as in Measure, Measure Space, and Probability Measure, 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 outer measure is the content of claim 1 of Existence of Lebesgue Measure on the Real Line.