What is Lebesgue measure? — Mathematics

Definition 9.10 University Mathematics — Year 3 · Chapter 9 — Measure Theory

The Lebesgue outer measure of $A \subseteq \R$ is

\lambda^*(A) = \inf\Bigl\{\sum_{n} (b_n - a_n) : A \subseteq \bigcup_n \intoo{a_n}{b_n}\Bigr\}

(countable covers by open intervals).

Examples

Example 9.14

The Cantor set (Exercise 6.10) has $\lambda(C) = 0$ : $C \subseteq C_n$ , a union of $2^n$ intervals of length $3^{-n}$ , so $\lambda(C) \leq (2/3)^n \to 0$ . An uncountable null set — cardinality does not see measure. Conversely, fat Cantor sets (Exercise 9.5) are nowhere dense with positive measure: topology does not see measure either. The weekend problem pushes this interplay to its striking conclusion: there are Lebesgue-measurable sets that are not Borel.

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