Lebesgue Measure

You now have all the pieces: outer measure builds a size function $\lambda^*$ on every subset of $\mathbb{R}$, and Carathéodory’s criterion isolates the sets on which $\lambda^*$ becomes a genuine, countably additive measure. Putting them together yields the Lebesgue measure — the canonical notion of length on $\mathbb{R}$.

The Lebesgue σ-algebra and Lebesgue measure

Definition. The Lebesgue σ-algebra $\mathcal{L}$ is the collection of all sets $E \subseteq \mathbb{R}$ that are Carathéodory-measurable with respect to the Lebesgue outer measure $\lambda^*$:

$$\mathcal{L} \;\coloneqq\; \{E \subseteq \mathbb{R} : \lambda^*(A) = \lambda^*(A \cap E) + \lambda^*(A \cap E^c) \text{ for all } A \subseteq \mathbb{R}\}. \tag{1}$$

The Lebesgue measure is the restriction

$$\lambda \;\coloneqq\; \lambda^*\restriction_{\mathcal{L}} \colon \mathcal{L} \to [0, +\infty]. \tag{2}$$

By Carathéodory’s theorem, $\mathcal{L}$ is a σ-algebra, and $\lambda$ is a complete, countably additive measure on $\mathcal{L}$.

Every open set is Lebesgue measurable

The first task is to verify that the Lebesgue measure captures ordinary geometry — that is, that every open interval, and hence every open set, is in $\mathcal{L}$.

Claim. Every open interval $(a, b)$ is in $\mathcal{L}$.

Proof. You need to show that for any test set $A$,

$$\lambda^*(A) \geq \lambda^*(A \cap (a,b)) + \lambda^*(A \cap (a,b)^c).$$

Fix a countable cover of $A$ by open intervals $\{I_k\}$ with $\sum_k |I_k| \leq \lambda^*(A) + \varepsilon$. For each $k$, split $I_k$ into $I_k \cap (a, b)$ and $I_k \setminus (a, b)$ (the latter being at most two intervals). These splits give covers of $A \cap (a, b)$ and $A \cap (a, b)^c$ respectively, with total lengths summing to $|I_k|$. Adding over $k$:

$$\lambda^*(A \cap (a,b)) + \lambda^*(A \cap (a,b)^c) \leq \sum_k |I_k| \leq \lambda^*(A) + \varepsilon.$$

Since $\varepsilon$ is arbitrary, the claim follows.

Because every open set in $\mathbb{R}$ is a countable union of disjoint open intervals (a fact from real analysis), and $\mathcal{L}$ is closed under countable unions, every open set is in $\mathcal{L}$. By taking complements, every closed set is also in $\mathcal{L}$.

The Borel sets are Lebesgue measurable

Since $\mathcal{L}$ contains all open sets and is a σ-algebra, it must contain the smallest σ-algebra generated by the open sets — the Borel σ-algebra $\mathcal{B}(\mathbb{R})$ from σ-Algebra:

$$\mathcal{B}(\mathbb{R}) \;\subseteq\; \mathcal{L}. \tag{3}$$

The inclusion is strict: $\mathcal{L}$ contains some non-Borel sets (the subsets of Borel null sets, which land in $\mathcal{L}$ by completeness but may not be Borel). In particular, every Borel set is Lebesgue measurable, but the Lebesgue σ-algebra is strictly larger than $\mathcal{B}(\mathbb{R})$.

Basic computations

Intervals

For any bounded interval $I$ with endpoints $a \leq b$,

$$\lambda(I) = b - a, \tag{4}$$

regardless of whether the endpoints are included. This is equation (5) from Outer Measure, applied to an element of $\mathcal{L}$.

For $\lambda(\{a\}) = 0$: a singleton can be covered by $(a - \varepsilon, a + \varepsilon)$ for any $\varepsilon > 0$, giving $\lambda^*(\{a\}) \leq 2\varepsilon$; hence $\lambda(\{a\}) = 0$. The formulas for open and closed intervals then follow from countable additivity.

Countable sets are null

A set $N$ is called a null set (or measure-zero set) when $\lambda(N) = 0$. Any countable set is a null set:

$$\lambda\!\left(\{x_1, x_2, x_3, \ldots\}\right) = \sum_{k=1}^{\infty} \lambda(\{x_k\}) = \sum_{k=1}^{\infty} 0 = 0. \tag{5}$$

This confirms the intuition from Introduction to Measure: $\lambda(\mathbb{Q} \cap [0,1]) = 0$, even though the rationals are dense in $[0,1]$.

Open and closed sets in $[0,1]$

The measure of an open set $U$ decomposes cleanly. Every open $U \subseteq [0,1]$ can be written as a countable disjoint union of open intervals $U = \bigsqcup_k (a_k, b_k)$, so

$$\lambda(U) = \sum_{k} (b_k - a_k). \tag{6}$$

Its closed complement $F = [0,1] \setminus U$ satisfies $\lambda(F) = 1 - \lambda(U)$, by the countable additivity of $\lambda$ on the disjoint decomposition $[0,1] = F \sqcup U$.

The Cantor set is a striking example. Start from $[0,1]$; at each stage remove the open middle third of every remaining interval. After countably many steps, what remains is a closed set $C$ with $\lambda(C) = 0$ (because the removed intervals account for total length $1/3 + 2/9 + 4/27 + \cdots = 1$). Yet $C$ is uncountable — it has the cardinality of $\mathbb{R}$.

Translation invariance

Theorem. For any $E \in \mathcal{L}$ and $t \in \mathbb{R}$, the translate $E + t \coloneqq \{x + t : x \in E\}$ satisfies $E + t \in \mathcal{L}$ and

$$\lambda(E + t) = \lambda(E). \tag{7}$$

Why. Translating a cover by intervals of $E$ by $t$ gives a cover of $E + t$ with the same total length. So $\lambda^*(E + t) = \lambda^*(E)$. The Carathéodory measurability of $E + t$ follows similarly.

Translation invariance $(7)$ is a defining geometric property of length: the size of a set does not depend on where it sits on the line. Together with countable additivity and the normalization $\lambda([0,1]) = 1$, this uniquely characterises Lebesgue measure among all measures on $\mathcal{B}(\mathbb{R})$ that are finite on bounded sets.

Summary

• The Lebesgue σ-algebra $\mathcal{L}$ (equation $(1)$) is the collection of Carathéodory-measurable sets for $\lambda^*$; the Lebesgue measure $\lambda$ (equation $(2)$) is $\lambda^*$ restricted to $\mathcal{L}$.
• $\lambda$ is a complete, countably additive measure, by Carathéodory’s theorem.
• Every open set, and therefore every Borel set, belongs to $\mathcal{L}$ — inclusion $(3)$. The Lebesgue σ-algebra is strictly larger than $\mathcal{B}(\mathbb{R})$.
• On intervals, $\lambda$ agrees with length: $\lambda(I) = b - a$ — equation $(4)$.
• Countable sets are null sets — equation $(5)$; in particular $\lambda(\mathbb{Q} \cap [0,1]) = 0$.
• The Cantor set is a closed, uncountable null set: a striking example of the subtlety of measure zero.
• Translation invariance $\lambda(E + t) = \lambda(E)$ — equation $(7)$ — reflects that length is a geometric (not positional) property of a set.