Lebesgue Measure

Motivation

The Lebesgue measure, denoted by $λ$ or $m$ , is the standard method for assigning a "size" (length, area, or volume) to a vast class of subsets in Euclidean space $R^{n}$ . It formalizes and extends the intuitive geometric concept of length beyond simple intervals to more complex sets. It serves as the foundational measure for the Lebesgue integral on $R^{n}$ .

Construction Outline

The measure is constructed from an outer measure $λ^{*}$ , which is defined for all subsets of $R^{n}$ .

Outer Measure $λ^{*}$ : For any set $A \subseteq R^{n}$ , its outer measure is the smallest possible total volume of a countable collection of open boxes that covers $A$ . In one dimension ( $R$ ), this is:

λ^{*} (A) = inf {\sum_{k = 1}^{\infty} ℓ (I_{k}) | A \subseteq ⋃_{k = 1}^{\infty} I_{k}, I_{k} are open intervals}

Here, $ℓ (I_{k})$ is the length of the interval $I_{k}$ . This process finds the most "efficient" cover for the set.

Measurable Sets (Carathéodory's Criterion): A set $E$ is declared Lebesgue measurable if it cleanly "splits" any other set $A \subseteq R^{n}$ in an additive manner:$$
\lambda^(A) = \lambda^(A \cap E) + \lambda^*(A \cap E^c) \quad \text{for all } A \subseteq \mathbb{R}^n
Lebesgue Measure $λ$ : The Lebesgue measure is simply the outer measure $λ^{*}$ when restricted to this $σ$ -algebra of measurable sets.
$λ (E) = λ^{*} (E) for any measurable set E \in L (R^{n})$

Properties

The Lebesgue measure on the measure space $(R^{n}, L (R^{n}), λ)$ has several fundamental properties:

Normalization: The measure of an interval $[a, b]$ is its length, $b - a$ . The measure of a box in $R^{n}$ is its volume.
Translation invariance: Shifting a set does not change its measure. For any measurable set $E$ and any vector $x \in R^{n}$ , we have $λ (E + x) = λ (E)$ .
Countable additivity: For any countable collection of disjoint measurable sets ${E_{k}}$ , the measure of their union is the sum of their measures: $λ (⋃_{k = 1}^{\infty} E_{k}) = \sum_{k = 1}^{\infty} λ (E_{k})$
Completeness: Any subset of a set with measure zero is itself measurable and also has measure zero.

Key Examples

The Lebesgue measure of any countable set is zero. This is why the set of rational numbers, $Q$ , has $λ (Q) = 0$ . This result is critical for understanding why the Lebesgue integral of functions like the Dirichlet function is zero.
All open sets, closed sets, and Borel sets are Lebesgue measurable. However, the collection of Lebesgue measurable sets is strictly larger than the Borel sets.
Non-measurable sets (like Vitali sets) exist. These are exotic sets that cannot be assigned a Lebesgue measure, proving that not every subset of $R^{n}$ is measurable. Their construction typically requires the Axiom of Choice.