Existence of Lebesgue Measure on the Real Line

Let $\lambda^{*}$ be the Lebesgue outer measure on the real line. Then:

1. $\lambda^{*}$ is an outer measure on $\mathbb{R}$;
2. every Borel set is Carathéodory measurable with respect to $\lambda^{*}$;
3. consequently, by Caratheodory Extension Theorem, the restriction $\lambda$ of $\lambda^{*}$ to $\mathcal{B}(\mathbb{R})$ is a measure, called Lebesgue measure on $\mathbb{R}$;
4. for all real $a\le b$,

and more generally every interval with endpoints $a\le b$ has Lebesgue measure $b-a$; 5. $\lambda$ is $\sigma$-finite.