Lebesgue Measure on the Real Line: Existence, Lengths of Intervals and Uniqueness

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $\lambda^{*}$ be the Lebesgue outer measure on $\mathbb{R}$, let $\mathcal{B}(\mathbb{R})$ be the Borel $\sigma$-algebra of $\mathbb{R}$, so that $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ is a measurable space by Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iii), and let $\lambda$ be Lebesgue measure on $\mathbb{R}$. Intervals are as in Intervals of the Real Line (ii).

(i) For all $a,b\in\mathbb{R}$ with $a\le b$, the intervals $(a,b)$, $[a,b]$, $[a,b)$ and $(a,b]$ belong to $\mathcal{B}(\mathbb{R})$ by Intervals Are Borel, and Rays, Half-Open Intervals and Rational Intervals Generate the Borel Sigma-Algebra of the Real Line (ii), and

$$\lambda\bigl((a,b)\bigr)=\lambda\bigl([a,b]\bigr)=\lambda\bigl([a,b)\bigr)=\lambda\bigl((a,b]\bigr)=b-a.$$

Assume moreover that the axiom of dependent choice holds.

(ii) $\lambda^{*}$ is an outer measure on $\mathbb{R}$.

(iii) Every Borel subset of $\mathbb{R}$ is Carathéodory measurable for $\lambda^{*}$, which is an outer measure by (ii); that is, $\mathcal{B}(\mathbb{R})\subseteq\mathcal{M}(\lambda^{*})$, where $\mathcal{M}(\lambda^{*})$ is the set of Carathéodory measurable sets for $\lambda^{*}$.

(iv) $\lambda$ is a measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$.

(v) $\lambda$ is $\sigma$-finite.

(vi) If $\nu$ is a measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ with $\nu\bigl((a,b]\bigr)=b-a$ for all $a,b\in\mathbb{R}$ with $a<b$, then $\nu(B)=\lambda(B)$ for every $B\in\mathcal{B}(\mathbb{R})$, that is, $\nu=\lambda$.