Lebesgue Outer Measure and Caratheodory Extension

An outer measure on $X$ is a function $\mu^* : \mathcal{P}(X) \to [0, \infty]$ satisfying:

1. $\mu^*(\varnothing) = 0$.
2. Monotonicity: $A \subseteq B$ implies $\mu^*(A) \leq \mu^*(B)$.
3. Countable subadditivity: $\mu^*\left(\bigcup A_n\right) \leq \sum \mu^*(A_n)$.

Proposition 3.1. Every outer measure is subadditive: $\mu^*(A \cup B) \leq \mu^*(A) + \mu^*(B)$.

Proof. Apply countable subadditivity with $A_1 = A$, $A_2 = B$, $A_n = \varnothing$ for $n \geq 3$. $\square$

The Lebesgue outer measure of $A \subseteq \mathbb{R}$ is

where the infimum is taken over all countable coverings of $A$ by open intervals.

Properties of $m^*$.

1. $m^*(\varnothing) = 0$ (this follows immediately: cover with no intervals).
2. $m^*$ is translation invariant: $m^*(A + t) = m^*(A)$ for all $t \in \mathbb{R}$.
3. $m^*$ is monotone: $A \subseteq B$ implies $m^*(A) \leq m^*(B)$.
4. $m^*$ is countably subadditive.

Proposition 3.2. For any interval $I = [a, b]$, we have $m^*(I) = b - a$.

Proof. Cover $I$ by the single open interval $(a - \varepsilon, b + \varepsilon)$ to get $m^*(I) \leq b - a + 2\varepsilon$, so $m^*(I) \leq b - a$.

For the reverse inequality, suppose for contradiction that some countable cover $\{(a_n, b_n)\}$ of $[a,b]$ satisfies $S = \sum_n (b_n - a_n) < b - a$. Write $S_N = \sum_{n \leq N}(b_n - a_n)$ and choose $N$ with $S - S_N < (b-a)/3$, so $S_N < 2(b-a)/3$.

Set $K = [a,b] \setminus \bigcup_{n \leq N}(a_n, b_n)$. This is closed and bounded, hence compact, and it is covered by the tail $\bigcup_{n > N}(a_n, b_n)$, so $m^*(K) \leq S - S_N < (b-a)/3$. The rest of $[a,b]$ lies in the first $N$ intervals, and $m^*$ is monotone, so

a contradiction. Hence every cover has total length at least $b - a$, and $m^*([a,b]) \geq b - a$. $\square$

The finite-subcover route does not work here, and it is worth seeing why, because this proof is usually where the first wrong turn is made. Compactness does give a finite subcover of $[a,b]$; but the length of that finite subcover is bounded by the length of the original cover, which is exactly the quantity under dispute. Compactness alone yields no lower bound on the cover’s length, because the remaining gap $K$ is covered by the tail, and the tail is precisely the part a finite subcover discards. The argument has to keep the tail and bound it separately, and that is what the two estimates above do.

A set $A \subseteq X$ is $\mu^*$-measurable (Caratheodory measurable) if for every $E \subseteq X$:

Since subadditivity always gives $\mu^*(E) \leq \mu^*(E \cap A) + \mu^*(E \cap A^c)$, the condition reduces to:

Lemma 3.3. The collection $\mathcal{M}$ of all $\mu^*$-measurable sets is an algebra.

Proof. We verify closure under complement and finite union.

• Complement: If $A \in \mathcal{M}$, then $A^c \in \math{M}$ by symmetry of the condition.
• Finite union: If $A, B \in \mathcal{M}$, then for any $E$: $$\mu^*(E) = \mu^*(E \cap A) + \mu^*(E \cap A^c)$$ Applying measurability of $B$ to $E \cap A$ and to $E \cap A^c$: $$\mu^*(E \cap A) = \mu^*(E \cap A \cap B) + \mu^*(E \cap A \cap B^c)$$ $$\mu^*(E \cap A^c) = \mu^*(E \cap A^c \cap B) + \mu^*(E \cap A^c \cap B^c)$$ Combining: $\mu^*(E) = \mu^*(E \cap (A \cup B)) + \mu^*(E \cap (A \cup B)^c)$, so $A \cup B \in \mathcal{M}$. $\square$

Theorem 3.4 (Caratheodory Extension Theorem). If $\mu^*$ is an outer measure on $X$, then $\mathcal{M}$ (the collection of $\mu^*$-measurable sets) is a $\sigma$-algebra, and $\mu = \mu^*|_{\mathcal{M}}$ is a complete measure.

Proof sketch. The hard part is showing $\sigma$-algebra closure (countable unions). One shows:

1. $\mathcal{M}$ is an algebra (Lemma 3.3).
2. $\mathcal{M}$ is closed under countable unions: use the fact that if $A_n \in \mathcal{M}$ are pairwise disjoint, then for any $E$: $$\mu^*\left(E \cap \bigcup_{n=1}^{N} A_n\right) = \sum_{n=1}^{N} \mu^*(E \cap A_n)$$ by induction. Letting $N \to \infty$ and using countable subadditivity gives countable additivity on $\mathcal{M}$.
3. Completeness: if $N \subseteq M$ with $\mu^*(M) = 0$, then $N \in \mathcal{M}$ (since $\mu^*(E \cap N) \leq \mu^*(N) = 0$). $\square$

Corollary 3.5. When $\mu^*$ is the Lebesgue outer measure on $\mathbb{R}$, the resulting $\sigma$-algebra $\mathcal{M}$ contains the Borel $\sigma$-algebra: $\mathcal{B}(\mathbb{R}) \subseteq \mathcal{M}$. The restriction $\mu = m^*|_{\mathcal{M}}$ is the Lebesgue measure.

Not every subset of $\mathbb{R}$ is Lebesgue measurable.

Theorem 3.6 (Vitali, 1905). Assuming the Axiom of Choice, there exists $V \subseteq [0, 1]$ that is not Lebesgue measurable.

Proof sketch. Define $x \sim y$ iff $x - y \in \mathbb{Q}$. This partitions $[0, 1]$ into equivalence classes. Choose one representative from each class to form $V$. If $V$ were measurable, then by translation invariance and countable additivity, $m^*(V)$ would have to be either $0$ or positive, both leading to contradictions when considering $\bigcup_{q \in \mathbb{Q} \cap [-1, 1]} (V + q) \subseteq [-1, 2]$. $\square$

Corollary 3.7. The existence of non-measurable sets implies that translation-invariant, countably additive measures on all subsets of $\mathbb{R}$ cannot exist. This is why we restrict to the $\sigma$-algebra $\mathcal{M}$ of measurable sets.

Definition 3.8. A Borel measure $\mu$ on $\mathbb{R}^n$ is:

1. Outer regular if $\mu(A) = \inf\{\mu(U) : A \subseteq U, U \text{ open}\}$ for all Borel sets $A$.
2. Inner regular if $\mu(A) = \sup\{\mu(K) : K \subseteq A, K \text{ compact}\}$ for all Borel sets $A$.

Theorem 3.9. Lebesgue measure on $\mathbb{R}^n$ is both outer and inner regular.

Proof sketch. Outer regularity follows from the definition of $m^*$ as an infimum over open coverings. Inner regularity follows from the fact that any measurable set can be approximated from within by compact sets (using the continuity of measure from above). $\square$

Theorem 3.10 (Lebesgue Density Theorem). For any Lebesgue measurable set $A \subseteq \mathbb{R}$, the density

exists and equals $1$ for almost every $x \in A$ and $0$ for almost every $x \notin A$.

Intuition. Most points of $A$ are “density points” where $A$ locally fills up most of the ball. This theorem is fundamental in geometric measure theory and has applications to differentiation of integrals.

1. Confusing outer measure with measure. The Lebesgue outer measure $m^*$ is defined on all subsets of $\mathbb{R}$, but it is not countably additive on all subsets. Only its restriction to $\mathcal{M}$ (the measurable sets) is a measure.

2. Assuming all sets are measurable. The Vitali construction shows that non-measurable sets exist (using the Axiom of Choice). In practice, all sets encountered in analysis are measurable.

3. Forgetting the infimum in the definition. The Lebesgue outer measure uses an infimum over all coverings, not a specific covering. Using a particular covering gives only an upper bound.

4. Confusing Caratheodory measurability with Borel measurability. Caratheodory measurability is defined using the outer measure splitting condition, while Borel measurability is defined using preimages of open sets. The two notions coincide for Lebesgue measure, but Caratheodory’s approach is more general.

5. Ignoring the role of the Axiom of Choice. The Vitali construction requires the Axiom of Choice to select representatives from equivalence classes. Without AC, it is consistent that all subsets of $\mathbb{R}$ are Lebesgue measurable.

Example 3.11. Compute $m^*((0, 1) \cup (2, 3))$.

Solution. Subadditivity gives $m^*((0,1) \cup (2,3)) \leq m^*((0,1)) + m^*((2,3)) = 2$.

For the reverse inequality, let $\{(a_n, b_n)\}$ be any countable cover of $(0,1) \cup (2,3)$ and let $N$ be chosen so that the tail has small total length, $\sum_{n > N}(b_n - a_n) < 1/2$. The set $K = ((0,1) \cup (2,3)) \setminus \bigcup_{n \leq N}(a_n, b_n)$ has $m^*(K) \leq \sum_{n>N}(b_n - a_n) < 1/2$, and it lies inside $(0,1) \cup (2,3)$ with each of $(0,1)$ and $(2,3)$ contributing at most $1$ by 3.2, so $m^*(K) \leq 2$. The part covered by the first $N$ intervals has measure at most $S_N = \sum_{n\leq N}(b_n-a_n)$. Hence

and letting $N \to \infty$ along the tail estimate forces $S = \sum_n (b_n-a_n) \geq 2$. Since every cover has total length at least $2$, $m^*((0,1)\cup(2,3)) = 2$. $\square$

Note the shape of the argument: it is Proposition 3.2’s tail estimate again, not a claim that a cover “must cover both intervals”. A cover is free to spend all of its length on one interval, so disjointness alone yields no lower bound; the lower bound comes from bounding the uncovered remainder.

Example 3.12. Show that the Cantor set $C$ has Lebesgue measure zero.

Solution. At stage $n$ of the construction, $C$ is contained in the union of the $2^n$ closed intervals remaining, each of length $3^{-n}$. Those intervals form an open cover of $C$ once widened by any $\varepsilon$, so

Since $m^*(C) \geq 0$, we get $m^*(C) = 0$.

The tempting wrong proof is to compute $m^*([0,1]) - \sum_{n} 2^{n-1} 3^{-n} = 1 - 1 = 0$ and conclude $m^*(C) = 0$. That subtracts outer measures of disjoint sets, and outer measure is subadditive, not additive: all it says is $\mu^*(A \cup B) \leq \mu^*(A) + \mu^*(B)$. There is no rule permitting $\mu^*(\text{whole}) - \mu^*(\text{part})$, and the argument would prove that every subset of $[0,1]$ has measure zero. The correct proof above never subtracts anything; it exhibits one cover after another whose total length tends to $0$, and the definition of $m^*$ is an infimum over covers.

Example 3.13. Prove that $m^*(A \cup B) = m^*(A) + m^*(B)$ when $d(A, B) > 0$.

Solution. When $d(A, B) = \inf\{|a - b| : a \in A, b \in B\} > 0$, there exist open sets $U \supseteq A$ and $V \supseteq B$ with $U \cap V = \varnothing$ (separation by open sets). Then $m^*(A \cup B) \leq m^*(U) + m^*(V)$ by subadditivity. For the reverse, any covering of $A \cup B$ can be split into coverings of $A$ and $B$ (using the separation), giving $m^*(A \cup B) \geq m^*(A) + m^*(B)$.

Example 3.14. The Lebesgue measure is translation invariant.

Proof. For any $A \subseteq \mathbb{R}$ and $t \in \mathbb{R}$, we have $m^*(A + t) = \inf\{\sum(b_n - a_n) : A + t \subseteq \bigcup(a_n, b_n)\}$. Substituting $u_n = a_n - t$, $v_n = b_n - t$, we get $m^*(A + t) = \inf\{\sum(v_n - u_n) : A \subseteq \bigcup(u_n, v_n)\} = m^*(A)$.

The Caratheodory extension theorem is the foundation for Lebesgue integration theory:

1. Measurable functions: A function $f: \mathbb{R} \to \mathbb{R}$ is Lebesgue measurable if $f^{-1}((a, \infty)) \in \mathcal{M}$ for all $a \in \mathbb{R}$.

2. Simple functions: Approximate measurable functions by step functions $\phi = \sum_{i=1}^{n} a_i \chi_{A_i}$ where $A_i \in \mathcal{M}$.

3. Lebesgue integral: $\int f \, dm = \sup\{\int \phi \, dm : 0 \leq \phi \leq f, \phi \text{ simple}\}$.

4. Monotone convergence theorem: If $0 \leq f_n \uparrow f$ pointwise, then $\int f_n \, dm \uparrow \int f \, dm$.

5. Dominated convergence theorem: If $f_n \to f$ pointwise and $|f_n| \leq g$ with $\int g \, dm < \infty$, then $\int f_n \, dm \to \int f \, dm$.

The Caratheodory construction can be applied to any outer measure on any set:

1. Abstract measure theory: Given an outer measure $\mu^*$ on $(X, \mathcal{P}(X))$, the Caratheodory measurable sets form a $\sigma$-algebra $\mathcal{M}$, and $\mu^*|_{\mathcal{M}}$ is a complete measure.

2. Hausdorff measures: On $\mathbb{R}^n$, the $s$-dimensional Hausdorff measure $\mathcal{H}^s$ is defined using coverings by sets of diameter at most $\delta$, taking the limit as $\delta \to 0$. For $s = n$, this recovers Lebesgue measure.

3. Haar measures: On locally compact groups, the Haar measure is a translation-invariant measure (unique up to scalar multiples). The Caratheodory construction is used in its construction.

4. Product measures: Given measures $\mu$ on $(X, \mathcal{A})$ and $\nu$ on $(Y, \mathcal{B})$, the product measure $\mu \times \nu$ on $(X \times Y, \mathcal{A} \otimes \mathcal{B})$ is constructed using the Caratheodory extension from rectangles.

Lemma 3.15. If $\mu^*$ is an outer measure and $A \subseteq B$, then $\mu^*(A) \leq \mu^*(B)$.

Proof. Any covering of $B$ is also a covering of $A$, so the infimum over coverings of $A$ is at most the infimum over coverings of $B$.

Lemma 3.16. For any sequence $\{A_n\}$ of sets, $m^*\left(\bigcup_{n=1}^{\infty} A_n\right) \leq \sum_{n=1}^{\infty} m^*(A_n)$.

Proof. For each $n$ and each $\varepsilon > 0$, choose a covering $\{(a_{n,k}, b_{n,k})\}_{k=1}^{\infty}$ of $A_n$ with $\sum_{k}(b_{n,k} - a_{n,k}) < m^*(A_n) + \varepsilon/2^n$. Then $\{(a_{n,k}, b_{n,k})\}_{n,k}$ is a covering of $\bigcup A_n$ with total length less than $\sum m^*(A_n) + \varepsilon$. Since $\varepsilon$ is arbitrary, the result follows.

Proposition 3.17. The Lebesgue outer measure of a countable set is zero.

Proof. Let $A = \{x_1, x_2, \ldots\}$. For each $\varepsilon > 0$, cover $x_n$ by $(x_n - \varepsilon/2^{n+1}, x_n + \varepsilon/2^{n+1})$. Then $m^*(A) \leq \sum_{n=1}^{\infty} \varepsilon/2^n = \varepsilon$. Since $\varepsilon$ is arbitrary, $m^*(A) = 0$.

Corollary 3.18. The rationals $\mathbb{Q} \cap [0, 1]$ have Lebesgue measure zero. More generally, any countable subset of $\mathbb{R}$ has measure zero.

Proposition 3.19. For any open set $U \subseteq \mathbb{R}$, $m^*(U)$ equals the sum of the lengths of its connected components.

Proof. An open subset of $\mathbb{R}$ is a countable disjoint union of open intervals, $U = \bigsqcup_{n=1}^{\infty}(a_n, b_n)$; countability of the components follows from the fact that each contains a rational. Subadditivity gives $m^*(U) \leq \sum_n (b_n - a_n)$.

For the reverse, let $\{(c_k, d_k)\}$ be any countable cover of $U$. Fix $n$. The tail $\sum_{k > K}(d_k - c_k)$ can be made smaller than any $\varepsilon$, so $(a_n, b_n) \setminus \bigcup_{k \leq K}(c_k, d_k)$ has outer measure at most $\varepsilon$ and is covered, in the limit, by an interval of length $\leq \varepsilon$; while $(a_n, b_n) \cap \bigcup_{k\leq K}(c_k, d_k)$ is covered by finitely many of them. Proposition 3.2’s estimate, applied to $(a_n, b_n)$, then gives $\sum_k (d_k - c_k) \geq b_n - a_n$. Taking $n$ to infinity, $\sum_k (d_k - c_k) \geq \sum_n (b_n - a_n)$.

The finiteness of the components is what licenses taking this limit, and it is why “any open set is a countable union of intervals” must be proved rather than assumed: a cover’s contribution to each component can be made finite.

Corollary 3.21. The Lebesgue measure of any open set is the sum of the lengths of its connected components. (This is now a restatement of Proposition 3.19: it was originally numbered to follow a separate statement of the same fact, which has been merged into it.)

1. Measure-theoretic probability: Lebesgue measure on $[0, 1]$ is the uniform probability measure. The Caratheodory construction underpins the construction of probability measures from cumulative distribution functions.

2. Geometric measure theory: Hausdorff measures (generalisations of Lebesgue measure to fractional dimensions) are constructed using the same Caratheodory method. They measure the “size” of fractals and lower-dimensional objects.

3. Harmonic analysis: The Lebesgue measure on $\mathbb{R}^n$ is the Haar measure for the additive group $(\mathbb{R}^n, +)$. Fourier analysis on locally compact groups relies on Haar measures.

4. Functional analysis: The Lebesgue spaces $L^p(\mathbb{R}^n)$ are defined using Lebesgue measure. These spaces are fundamental in PDE theory and functional analysis.

• Halmos, P.R. Measure Theory. Springer, 1974. Chapter 3 covers outer measures and the Caratheodory construction in detail.
• Folland, G.B. Real Analysis: Modern Techniques and Their Applications. Wiley, 1999. Chapter 2 provides a modern treatment of measure theory.
• Royden, H.L. and Fitzpatrick, P.M. Real Analysis. Pearson, 2010. Chapter 3 covers Lebesgue measure and integration.
• Stein, E.M. and Shakarchi, R. Real Analysis: Measure Theory, Integration, and Hilbert Spaces. Princeton University Press, 2005. Chapter 1 introduces measure theory via the Lebesgue construction.

1. Continuum Hypothesis and measurability: It is consistent with ZFC that all subsets of $\mathbb{R}$ are Lebesgue measurable (Solovay, 1970), but this requires the assumption that an inaccessible cardinal exists.

2. Regular measures on non-separable spaces: The regularity properties of Lebesgue measure may fail for measures on non-separable measure spaces.

3. Which outer measures arise from Lebesgue outer measure in higher dimensions? Hausdorff’s construction gives $\mathcal{H}^s$ on $\mathbb{R}^n$ for every real $s$, and it is not known in general which of these agree with each other off the integer dimensions.

The Lebesgue outer measure and Caratheodory extension theorem provide the foundation for modern measure theory. The key ideas are:

1. Outer measures extend the notion of “size” to all subsets, but may lack countable additivity.
2. Caratheodory measurability identifies the subsets where the outer measure behaves like a measure.
3. The extension theorem guarantees that the restriction to measurable sets is a complete measure.
4. Non-measurable sets exist (requiring the Axiom of Choice), but are “pathological” and rarely encountered in practice.
5. The construction generalises to Hausdorff measures, Haar measures, and product measures.

• Lebesgue (1902): Introduced the Lebesgue integral and measure in his doctoral thesis.
• Caratheodory (1914): Generalised the construction to arbitrary outer measures.
• Vitali (1905): Proved the existence of non-measurable sets using the Axiom of Choice.
• Banach (1923): Showed that a translation-invariant, finitely additive measure on all subsets of $\mathbb{R}$ exists if we drop countable additivity (using a weaker form of the Axiom of Choice).

• Lebesgue Measurable Sets and Non-Measurable Sets: Further exploration of which subsets of $\mathbb{R}$ are measurable and the structure of the $\sigma$-algebra of measurable sets.

• Measurable Functions: Builds on the measure-theoretic foundation by defining functions that respect the $\sigma$-algebra structure established here.

• $L^p$ Spaces: Defines Banach spaces of measurable functions with finite $p$-norm, relying on Lebesgue measure constructed via the Caratheodory extension.

• Radon-Nikodym Derivative and Lebesgue Decomposition: Uses absolute continuity and decomposition concepts that depend on the measure theory developed here.

• Quantum Mechanics

• Graph Theory

• Classical Mechanics

• Electromagnetism

• Statistical Learning

• Statistical Mechanics

• $L^p$ Spaces
• Fubini and Tonelli Theorems
• Lebesgue Integration
• Lebesgue Measurable Sets and Non-Measurable Sets