Lebesgue Measure - Core Definitions

Lebesgue measure is the most important measure in analysis, providing a rigorous foundation for integration on Euclidean spaces. It extends the intuitive notion of length, area, and volume to a vast class of sets.

DefinitionOuter Measure

The Lebesgue outer measure on $\mathbb{R}$ is the function $\lambda^*: \mathcal{P}(\mathbb{R}) \to [0, \infty]$ defined by: $\lambda^*(E) = \inf \left\{\sum_{n=1}^{\infty} \ell(I_n) : E \subseteq \bigcup_{n=1}^{\infty} I_n, I_n \text{ are intervals}\right\}$ where $\ell(I)$ denotes the length of interval $I$ .

The outer measure assigns to each set the infimum of the total length of countable interval covers.

The outer measure satisfies several important properties but is not quite a measure on all of $\mathcal{P}(\mathbb{R})$ because it may fail countable additivity for arbitrary collections of sets. This leads to the concept of measurability.

DefinitionLebesgue Measurable Sets

A set $E \subseteq \mathbb{R}$ is Lebesgue measurable if for every set $A \subseteq \mathbb{R}$ : $\lambda^*(A) = \lambda^*(A \cap E) + \lambda^*(A \cap E^c)$

This is the Caratheodory criterion for measurability. The collection of all Lebesgue measurable sets forms a sigma-algebra $\mathcal{L}$ , called the Lebesgue sigma-algebra.

ExampleMeasurable Sets

The following sets are Lebesgue measurable:

All intervals (open, closed, half-open)
All Borel sets (elements of $\mathcal{B}(\mathbb{R})$ )
All countable sets (with measure zero)
Complements and countable unions of measurable sets

In fact, the Borel sigma-algebra $\mathcal{B}(\mathbb{R}) \subsetneq \mathcal{L}$ , meaning there exist Lebesgue measurable sets that are not Borel sets.

DefinitionLebesgue Measure

The Lebesgue measure $\lambda$ on $\mathbb{R}$ is the restriction of the outer measure $\lambda^*$ to the Lebesgue sigma-algebra $\mathcal{L}$ : $\lambda(E) = \lambda^*(E) \text{ for all } E \in \mathcal{L}$

This restriction is a complete measure on $(\mathbb{R}, \mathcal{L})$ .

Remark

Lebesgue measure satisfies translation invariance: for any $a \in \mathbb{R}$ and measurable set $E$ , the translated set $E + a = \{x + a : x \in E\}$ is measurable and $\lambda(E + a) = \lambda(E)$ . This property distinguishes Lebesgue measure among all measures on $\mathbb{R}$ .