Lebesgue Measure#

Definition: Length of an Interval

Let \(I\) be an interval of \(\mathbb{R}\).

The length of \(I\) is an extended real number defined as:

\(b - a\) if \(I\) has one of the forms \([a;b]\), \((a;b)\), \([a;b)\) or \((a;b]\), where \(a, b \in \mathbb{R}\);
\(\infty\) if \(I\) has one of the forms \([a;\infty)\), \((a; \infty)\), \((-\infty; b]\), \((-\infty; b)\) or \((-\infty;+\infty)\), where \(a, b \in \mathbb{R}\).

Notation

\[ l(I) \]

Definition: Generalized Volume of a Cuboid

Let \(C\) be an open cuboid in the Euclidean space \(\mathbb{R}^n\) with sides \(I_1, \dotsc, I_n\).

The volume of \(C\) is the product of the lengths of \(I_1, \dotsc, I_n\):

\[ \prod_{k = 1}^n l(I_k) \]

Notation

\[ \mathop{\operatorname{vol}}(C) \]

Theorem: Lebesgue Outer Measure

The function \(\lambda^{\ast}: \mathcal{P}(\mathbb{R}^n) \to \overline{\mathbb{R}}\) from the powerset of the Euclidean space \(\mathbb{R}^n\) to the extended real numbers \(\overline{\mathbb{R}}\) which is defined for each \(E \subseteq \mathbb{R}^n\) as the infimum of the total volumes of all sequences \((C_k)_{k \in \mathbb{N}}\) of open cuboids which cover \(E\), i.e.

\[ \lambda^{\ast}(E) \overset{\text{def}}{=} \inf \left\{\sum_{k = 1}^{\infty} \mathop{\operatorname{vol}}(C_k) : (C_k)_{k \in \mathbb{N}} \text{ is a sequence of open cuboids and } E \subseteq \bigcup_{k=1}^{\infty} C_k\right\}, \]

is an outer measure on \(\mathbb{R}^n\).

Proof

TODO

Definition: Lebesgue Outer Measure

We call \(\lambda^{\ast}\) the Lebesgue outer measure on \(\mathbb{R}^n\).

Definition: Lebesgue Measure

Let \(\lambda^{\ast}\) be the Lebesgue outer measure on the Euclidean space \(\mathbb{R}^n\) and let \(\Sigma\) be the collection of all \(\lambda^{\ast}\)-measurable sets.

The Lebesgue measure on \(\mathbb{R}^n\) is the measure \(\lambda^n: \Sigma \to \overline{\mathbb{R}}\) generated by Carathéodory's extension theorem using the Lebesgue outer measure \(\lambda^{\ast}\).

Definition: Lebesgue-Measurability

A subset \(S \subseteq \mathbb{R}^n\) is Lebesgue-measurable if \(S \in \Sigma\).

The Lebesgue measure is necessary for defining integration for real-valued functions \(f: \mathcal{D} \subseteq X \to \mathbb{R}\) but \(\mathcal{D}\) itself may have nothing to do with \(\mathbb{R}^n\) or \(\mathbb{R}\).