Outer Measure — Mathematics Knowledge System

Overview

An outer measure is a set function defined on all subsets of a space (not just measurable ones) that satisfies monotonicity and countable sub-additivity. Outer measures are the starting point for Carathéodory's construction: from any outer measure one extracts a sigma-algebra of 'measurable' sets on which the outer measure restricts to a genuine measure. This machinery produces Lebesgue measure, Hausdorff measure, and many others.

Intuition

Think of outer measure as a 'pessimistic cover': mu*(E) is the smallest total length (or volume) of countably many simple covering sets needed to cover E from the outside. You cover every set, but only 'measurable' sets — those that split every test set additively — earn the right to be called measurable. The outer measure of a measurable set equals its measure.

Formal Definition

Definition

An outer measure on a set X is a function mu*: P(X) -> [0, inf] satisfying: (1) mu*(empty) = 0, (2) monotonicity: A ⊆ B implies mu*(A) <= mu*(B), (3) countable sub-additivity: mu*(union A_n) <= sum mu*(A_n). The standard construction: given a collection E of 'elementary' sets with a pre-measure rho, define mu*(A) = inf { sum rho(E_n) : A ⊆ union E_n, each E_n in E }.

\mu^*(\emptyset) = 0

Null empty set

A \subseteq B \Rightarrow \mu^*(A) \leq \mu^*(B)

Monotonicity

\mu^*\!\left(\bigcup_{n=1}^{\infty} A_n\right) \leq \sum_{n=1}^{\infty} \mu^*(A_n)

Countable sub-additivity

\mu^*(A) = \inf\!\left\{ \sum_{n=1}^{\infty} \rho(E_n) : A \subseteq \bigcup_{n=1}^{\infty} E_n,\; E_n \in \mathcal{E} \right\}

Outer measure from a pre-measure

Notation

Notation	Meaning
$\mu^*$	Outer measure
$\lambda^*$	Lebesgue outer measure on R^n

Properties

Not necessarily additive

\mu^* \text{ need not be additive on disjoint sets; only sub-additivity is guaranteed.}

Countable sets have Lebesgue outer measure zero

\text{For Lebesgue outer measure } \lambda^* \text{ on } \mathbb{R},\; \lambda^*(\{x_1, x_2, \ldots\}) = 0.

Outer measure of an interval

\lambda^*([a,b]) = \lambda^*((a,b)) = b - a \quad (a \leq b)

Theorems

Theorem 1: Carathéodory measurability criterion

A set E \subseteq X is \mu^*\text{-measurable (in the sense of Carathéodory) if for every } A \subseteq X:\; \mu^*(A) = \mu^*(A \cap E) + \mu^*(A \cap E^c).

Theorem 2: Carathéodory's theorem

The collection \mathcal{M}_{\mu^*} \text{ of all } \mu^*\text{-measurable sets is a sigma-algebra, and } \mu^*|_{\mathcal{M}_{\mu^*}} \text{ is a complete measure.}

Worked Examples

E is countable: E = {q_1, q_2, ...}. For any epsilon > 0, cover q_k with the open interval (q_k - epsilon/2^{k+1}, q_k + epsilon/2^{k+1}).
$E \subseteq \bigcup_{k=1}^{\infty} \left(q_k - \frac{\varepsilon}{2^{k+1}},\, q_k + \frac{\varepsilon}{2^{k+1}}\right)$
The total length of the cover is sum_{k=1}^infty epsilon/2^k = epsilon.
$\sum_{k=1}^{\infty} \frac{\varepsilon}{2^k} = \varepsilon$
Since epsilon > 0 is arbitrary, lambda*(E) = 0.
$\lambda^*(E) = 0$

Answer: lambda*(Q ∩ [0,1]) = 0.

Practice Problems

Difficulty 6/10

Prove that the Lebesgue outer measure of a single point {x} in R is zero.

Difficulty 7/10

Show that the Lebesgue outer measure is translation-invariant: lambda*(E + t) = lambda*(E) for all E ⊆ R and t in R.

Difficulty 6/10

Which property distinguishes an outer measure from a full measure?

An outer measure is only defined on a sigma-algebraAn outer measure satisfies countable sub-additivity rather than countable additivityAn outer measure must be finiteAn outer measure cannot assign zero to any nonempty set

Common Mistakes

Common Mistake

Outer measure is countably additive on all disjoint sets.

Outer measure is only countably sub-additive in general. Countable additivity holds only on the sigma-algebra of Carathéodory-measurable sets.

Common Mistake

Every set is Carathéodory-measurable with respect to Lebesgue outer measure.

Assuming the axiom of choice, there exist non-measurable sets (e.g. Vitali sets) that fail Carathéodory's criterion.

Common Mistake

Outer measure of a union equals the sum of outer measures.

Only mu*(union A_n) <= sum mu*(A_n) is guaranteed (sub-additivity). Equality holds only for disjoint measurable sets.

Quiz

Historical Background

Constantin Carathéodory introduced the outer measure framework in 1914 as a unified way to construct measures, subsuming Lebesgue's earlier ad hoc construction. The approach is now standard because it separates the approximation step (defining the outer measure on all sets) from the measurability step (identifying well-behaved sets).

1902
Lebesgue defines outer measure on R implicitly in his thesis
Henri Lebesgue
1914
Carathéodory axiomatises outer measure and gives the general measurability criterion
Constantin Carathéodory

Summary

An outer measure mu*: P(X) -> [0,inf] satisfies mu*(empty) = 0, monotonicity, and countable sub-additivity.
The standard construction builds an outer measure by taking the infimum of total lengths of countable covers by elementary sets.
Carathéodory's criterion identifies the measurable sets: those that split every test set additively.
The measurable sets form a complete sigma-algebra on which the outer measure is a genuine measure.
Lebesgue measure on R^n is constructed via Carathéodory's theorem from the Lebesgue outer measure.

References

BookFolland, G. B. (1999). Real Analysis: Modern Techniques and Their Applications (2nd ed.). Wiley.
BookHalmos, P. R. (1950). Measure Theory. Van Nostrand.
WebsiteWikipedia — Outer measure