Lebesgue Measure - Key Properties

Lebesgue measure has several remarkable properties that make it the preferred measure for analysis on Euclidean spaces. These properties reflect both its geometric intuition and its analytical power.

DefinitionTranslation Invariance

Lebesgue measure is translation invariant: for any measurable set $E \subseteq \mathbb{R}$ and any $a \in \mathbb{R}$ : $\lambda(E + a) = \lambda(E)$ where $E + a = \{x + a : x \in E\}$ .

This property captures the geometric intuition that translating a set does not change its size.

ExampleNon-Measurable Sets

Surprisingly, not all subsets of $\mathbb{R}$ are Lebesgue measurable. The Vitali set provides a classical counterexample.

Define an equivalence relation on $[0,1]$ by $x \sim y$ if $x - y \in \mathbb{Q}$ . By the Axiom of Choice, we can select one representative from each equivalence class to form a set $V \subseteq [0,1]$ .

If $V$ were measurable, translation invariance would imply $\lambda(V + r) = \lambda(V)$ for all $r \in \mathbb{Q} \cap [-1,1]$ . But the sets $V + r$ are disjoint and their union contains $[0,1]$ , leading to a contradiction whether $\lambda(V) = 0$ or $\lambda(V) > 0$ .

Remark

Key properties of Lebesgue measure include:

Regularity: For any measurable set $E$ , there exist a $G_\delta$ set $G$ and an $F_\sigma$ set $F$ such that $F \subseteq E \subseteq G$ and $\lambda(G \setminus F) = 0$ .
Completeness: If $\lambda(E) = 0$ and $A \subseteq E$ , then $A$ is measurable with $\lambda(A) = 0$ .
$\sigma$ -finiteness: $\mathbb{R} = \bigcup_{n=1}^{\infty} [-n, n]$ where each $[-n, n]$ has finite measure.
Borel regularity: For any measurable $E$ , $\lambda(E) = \inf\{\lambda(U) : E \subseteq U, U \text{ open}\} = \sup\{\lambda(K) : K \subseteq E, K \text{ compact}\}$

These properties make Lebesgue measure uniquely suited for integration theory. The completeness property ensures that subsets of measure-zero sets are measurable, while regularity allows approximation by simpler sets. Translation invariance is essential for harmonic analysis and convolution operations.