Lebesgue Measure - Examples and Constructions

The construction of Lebesgue measure extends naturally to higher dimensions, providing a powerful framework for multivariable analysis and geometry.

DefinitionLebesgue Measure on $\mathbb{R}^n$

For $\mathbb{R}^n$ , we define Lebesgue measure by first considering rectangles. An $n$ -dimensional rectangle is a set of the form: $R = [a_1, b_1] \times [a_2, b_2] \times \cdots \times [a_n, b_n]$

The volume of $R$ is $\text{vol}(R) = \prod_{i=1}^{n} (b_i - a_i)$ .

The outer measure on $\mathbb{R}^n$ is defined as: $\lambda^*(E) = \inf \left\{\sum_{k=1}^{\infty} \text{vol}(R_k) : E \subseteq \bigcup_{k=1}^{\infty} R_k\right\}$

Lebesgue measure on $\mathbb{R}^n$ is then obtained by restricting $\lambda^*$ to the sigma-algebra of measurable sets.

ExampleComputing Lebesgue Measure

Here are several examples illustrating Lebesgue measure:

Intervals: For $a < b$ , $\lambda([a,b]) = \lambda((a,b)) = \lambda([a,b)) = \lambda((a,b]) = b - a$
Countable sets: Any countable set $E$ has $\lambda(E) = 0$ . In particular, $\lambda(\mathbb{Q}) = 0$ and $\lambda(\mathbb{Z}) = 0$ .
Cantor set: The standard middle-thirds Cantor set $C$ is uncountable but has $\lambda(C) = 0$ . It is obtained by iteratively removing middle thirds from $[0,1]$ .
Unit cube in $\mathbb{R}^n$ : $\lambda([0,1]^n) = 1$

Remark

An important construction technique is the method of approximation. To show a set $E$ is measurable with measure $m$ , it often suffices to find:

Open sets $U_n \supseteq E$ with $\lambda^*(U_n) \to m$
Closed sets $F_n \subseteq E$ with $\lambda(F_n) \to m$

If both limits agree, then $E$ is measurable with $\lambda(E) = m$ .

The relationship between Lebesgue measure and topology is deep. Every open set in $\mathbb{R}^n$ can be written as a countable union of almost disjoint cubes, and this decomposition is fundamental for analysis. The Lebesgue Differentiation Theorem, which states that for locally integrable functions $f$ , $\lim_{r \to 0} \frac{1}{\lambda(B_r(x))} \int_{B_r(x)} f \, d\lambda = f(x) \text{ for a.e. } x$ relies heavily on the geometric properties of Lebesgue measure, where $B_r(x)$ denotes the ball of radius $r$ centered at $x$ .