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

The axiom of dependent choice is used only in the proof of (ii), through Properties of Lebesgue Outer Measure on the Real Line (vii); every other step below is choice-free, given (ii), and the proof of (i), stated before the assumption of dependent choice, uses neither it nor (ii).

Proof of (ii). Since the axiom of dependent choice is assumed, $\lambda^{*}$ is an outer measure on $\mathbb{R}$ by Properties of Lebesgue Outer Measure on the Real Line (vii).

Proof of (iii). By (ii), $\lambda^{*}$ is an outer measure on $\mathbb{R}$, so Carathéodory's Theorem: the Measurable Sets of an Outer Measure Form a σ-Algebra on Which It Is a Measure (i) and Carathéodory's Theorem: the Measurable Sets of an Outer Measure Form a σ-Algebra on Which It Is a Measure (ii) apply to it with $X=\mathbb{R}$; write $\mathcal{M}=\mathcal{M}(\lambda^{*})$. Let $c\in\mathbb{R}$, and let $E=(c,\infty)$ be the ray of Intervals of the Real Line (iii), a subset of $\mathbb{R}$. For every $A\subseteq\mathbb{R}$, and so in particular for every $A\subseteq\mathbb{R}$ with $\lambda^{*}(A)<\infty$,

$$\lambda^{*}(A\cap E)+\lambda^{*}(A\setminus E)\le\lambda^{*}(A)$$

by Properties of Lebesgue Outer Measure on the Real Line (v). Hence $E\in\mathcal{M}$ by Carathéodory's Theorem: the Measurable Sets of an Outer Measure Form a σ-Algebra on Which It Is a Measure (i). So the set $\mathcal{R}=\{(c,\infty):c\in\mathbb{R}\}$, a subset of $\mathcal{P}(\mathbb{R})$, satisfies $\mathcal{R}\subseteq\mathcal{M}$. By Carathéodory's Theorem: the Measurable Sets of an Outer Measure Form a σ-Algebra on Which It Is a Measure (ii), $\mathcal{M}$ is a $\sigma$-algebra on $\mathbb{R}$, so the $\sigma$-algebra generated by $\mathcal{R}$ satisfies $\sigma(\mathcal{R})\subseteq\mathcal{M}$ by Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iv). Since $\mathcal{B}(\mathbb{R})=\sigma(\mathcal{R})$ by Intervals Are Borel, and Rays, Half-Open Intervals and Rational Intervals Generate the Borel Sigma-Algebra of the Real Line (iii), we get $\mathcal{B}(\mathbb{R})\subseteq\mathcal{M}$; as the elements of $\mathcal{M}$ are exactly the $\lambda^{*}$-measurable subsets of $\mathbb{R}$ by Carathéodory Measurable Sets of an Outer Measure (ii), every Borel subset of $\mathbb{R}$ is Carathéodory measurable for $\lambda^{*}$.

Proof of (iv). Let $\mu=\lambda^{*}|_{\mathcal{M}}$ be the restriction of $\lambda^{*}$ to $\mathcal{M}$; by Carathéodory's Theorem: the Measurable Sets of an Outer Measure Form a σ-Algebra on Which It Is a Measure (iv) it is a measure on $(\mathbb{R},\mathcal{M})$. For every $B\in\mathcal{B}(\mathbb{R})$ we have $B\in\mathcal{M}$ by (iii), so

$$\lambda(B)=\lambda^{*}(B)=\mu(B),$$

the first equality by Lebesgue Measure on the Real Line (i) and the second by Domains and Restrictions of Maps (iii). By Lebesgue Measure on the Real Line (i), $\lambda$ is a map $\mathcal{B}(\mathbb{R})\to[0,\infty]$. First, $\emptyset\in\mathcal{B}(\mathbb{R})$ by Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (vi), so $\lambda(\emptyset)=\mu(\emptyset)=0$ by Measure, Measure Space, and Probability Measure (i) applied to $\mu$. Next, let $(A_{m})$ be a pairwise disjoint sequence in $\mathcal{B}(\mathbb{R})$. As $\mathcal{B}(\mathbb{R})\subseteq\mathcal{M}$ ((iii)), $(A_{m})$ is also a pairwise disjoint sequence in $\mathcal{M}$ (Sequences and Maps of Sets: Complements, Traces, Two-Term Sequences, the Disjointing Sequence, and Sequences in a Subset (v)). Its union $U=\bigcup_{m\in\mathbb{N}}A_{m}$ lies in $\mathcal{B}(\mathbb{R})$ by σ-Algebra and Measurable Space (i), and countable additivity of $\mu$ (Measure, Measure Space, and Probability Measure (i)) gives $\mu(U)=\sum_{m\in\mathbb{N}}\mu(A_{m})$. As $\lambda(U)=\mu(U)$ and $\lambda(A_{m})=\mu(A_{m})$ for every $m\in\mathbb{N}$, the two series have the same terms, and therefore

$$\lambda\Bigl(\bigcup_{m\in\mathbb{N}}A_{m}\Bigr)=\sum_{m\in\mathbb{N}}\lambda(A_{m}).$$

So $\lambda$ is a measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ by Measure, Measure Space, and Probability Measure (i).

Proof of (i). Let $a,b\in\mathbb{R}$ with $a\le b$, and let $I$ be one of $(a,b)$, $[a,b]$, $[a,b)$, $(a,b]$. Then $I\in\mathcal{B}(\mathbb{R})$ by Intervals Are Borel, and Rays, Half-Open Intervals and Rational Intervals Generate the Borel Sigma-Algebra of the Real Line (ii), so $\lambda(I)=\lambda^{*}(I)$ by Lebesgue Measure on the Real Line (i), and $\lambda^{*}(I)=b-a$ by Properties of Lebesgue Outer Measure on the Real Line (iv).

Proof of (v). For $m\in\mathbb{N}$ we have $0<m$ in $\mathbb{R}$ by Inequalities in an Ordered Field: Mixed Transitivity, Strict Sums, Signs, Products, Natural Numbers, Halving, Reciprocals, Absolute Values and Squares (vi), hence $-m<m$. By Intervals of the Real Line as Defined Set Symbols, Typed Terms and Values of Maps (iii), applied with $D=\mathbb{N}$ and the maps $m\mapsto-m$ and $m\mapsto m$, there is exactly one map $m\mapsto X_{m}$ from $\mathbb{N}$ to $\mathcal{P}(\mathbb{R})$ with $X_{m}=(-m,m]$ for every $m\in\mathbb{N}$. Each $X_{m}$ lies in $\mathcal{B}(\mathbb{R})$ by Intervals Are Borel, and Rays, Half-Open Intervals and Rational Intervals Generate the Borel Sigma-Algebra of the Real Line (ii), so $(X_{m})$ is a sequence in $\mathcal{B}(\mathbb{R})$ (Sequences and Maps of Sets: Complements, Traces, Two-Term Sequences, the Disjointing Sequence, and Sequences in a Subset (v)).

We show $\bigcup_{m\in\mathbb{N}}X_{m}=\mathbb{R}$, with the union of Indexed Families of Sets and Their Union, Intersection and Product (ii). Every element of the union lies in some $X_{m}\subseteq\mathbb{R}$, so lies in $\mathbb{R}$. Conversely, let $x\in\mathbb{R}$. By The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (ix), applied to $|x|$, there is $n\in\mathbb{N}$ with $|x|<n$; then $-n<x<n$ by Inequalities in an Ordered Field: Mixed Transitivity, Strict Sums, Signs, Products, Natural Numbers, Halving, Reciprocals, Absolute Values and Squares (ix), so $x\in(-n,n]=X_{n}$, and $x$ lies in the union by Indexed Families of Sets and Their Union, Intersection and Product (ii).

Finally, for $m\in\mathbb{N}$, (i) with $a=-m$ and $b=m$ gives $\lambda(X_{m})=m-(-m)$, a real number, so $\lambda(X_{m})<\infty$ by Small Facts about [0,∞]: Zero, Finiteness and Sums of Zeros (ii). Hence $\lambda$ is $\sigma$-finite by Measure, Measure Space, and Probability Measure (iv).

Proof of (vi). Let $\nu$ be as in (vi), and let $\mathcal{E}=\{\emptyset\}\cup\{(a,b]:a,b\in\mathbb{R},\ a<b\}$ be the set of Intervals Are Borel, and Rays, Half-Open Intervals and Rational Intervals Generate the Borel Sigma-Algebra of the Real Line (v). It is a $\pi$-system on $\mathbb{R}$ by Intervals Are Borel, and Rays, Half-Open Intervals and Rational Intervals Generate the Borel Sigma-Algebra of the Real Line (v), and $\sigma(\mathcal{E})=\mathcal{B}(\mathbb{R})$ by Intervals Are Borel, and Rays, Half-Open Intervals and Rational Intervals Generate the Borel Sigma-Algebra of the Real Line (iv). The measures $\lambda$ (a measure by (iv)) and $\nu$ agree on $\mathcal{E}$: $\lambda(\emptyset)=0=\nu(\emptyset)$ by Measure, Measure Space, and Probability Measure (i), and for $a,b\in\mathbb{R}$ with $a<b$, $\lambda((a,b])=b-a$ by (i) and $\nu((a,b])=b-a$ by hypothesis. Let $(X_{m})$ be the sequence of the proof of (v). As $-m<m$, each $X_{m}=(-m,m]$ lies in $\mathcal{E}$, and $\mathcal{E}\subseteq\mathcal{P}(\mathbb{R})$, so $(X_{m})$ is a sequence in $\mathcal{E}$ (Sequences and Maps of Sets: Complements, Traces, Two-Term Sequences, the Disjointing Sequence, and Sequences in a Subset (v)); its union is $\mathbb{R}$ and $\lambda(X_{m})<\infty$ for every $m\in\mathbb{N}$, both shown there. By Uniqueness of Measures Agreeing on a Generating π-System (ii), applied to $(\mathbb{R},\mathcal{B}(\mathbb{R}))$, $\mathcal{E}$, $\mu=\lambda$, $\nu$ and $E_{m}=X_{m}$, we get $\nu(B)=\lambda(B)$ for every $B\in\mathcal{B}(\mathbb{R})$. As $\nu$ and $\lambda$ are both maps $\mathcal{B}(\mathbb{R})\to[0,\infty]$ by Measure, Measure Space, and Probability Measure (i), $\nu=\lambda$ by Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (iv). $\blacksquare$