Outer Measure

You want a function that assigns a “size” to every subset of $\mathbb{R}$. The σ-algebra told you which sets will ultimately be measurable, but before restricting to them you first build a preliminary size function — the outer measure — that is defined on all subsets. Its key virtue is that it is always well-defined; its key limitation is that it is only sub-additive, not fully additive. Fixing that limitation is exactly Carathéodory’s job in the next checkpoint.

The definition

The idea is to approximate the size of a set $E$ from the outside by covering it with open intervals and summing up their lengths. Taking the infimum over all countable covers gives the smallest possible total length needed to cover $E$.

Definition. For any set $E \subseteq \mathbb{R}$, the Lebesgue outer measure of $E$ is

$$\lambda^*(E) \;\coloneqq\; \inf\!\left\{ \sum_{k=1}^{\infty} (b_k - a_k) \;\middle|\; E \subseteq \bigcup_{k=1}^{\infty} (a_k, b_k) \right\}, \tag{1}$$

where the infimum is over all countable collections of open intervals $(a_k, b_k)$ covering $E$. If no finite bound exists, set $\lambda^*(E) = +\infty$.

A few notational notes:

• The intervals $(a_k, b_k)$ may overlap; $b_k - a_k$ is the length of the $k$-th interval.
• You are allowed to use the same interval more than once (though doing so only wastes coverage).
• A finite cover is a special case of a countable cover (pad with empty intervals).

The same construction works on $\mathbb{R}^n$ using $n$-dimensional boxes, or on any metric space using appropriate “elementary sets”, but for now you work in $\mathbb{R}$.

Three fundamental properties

(i) Outer measure of the empty set is zero

$$\lambda^*(\emptyset) = 0. \tag{2}$$

For any $\varepsilon > 0$, cover $\emptyset$ with a single interval $(0, \varepsilon)$ of length $\varepsilon$. Taking $\varepsilon \to 0$ gives $\lambda^*(\emptyset) \leq 0$; since lengths are non-negative, $\lambda^*(\emptyset) = 0$.

(ii) Monotonicity

If $A \subseteq B \subseteq \mathbb{R}$, then

$$\lambda^*(A) \leq \lambda^*(B). \tag{3}$$

Every countable cover of $B$ is also a countable cover of $A$. Taking the infimum over covers of $B$ therefore gives a number that is already at least as large as $\lambda^*(A)$.

(iii) Countable sub-additivity

For any sequence of sets $E_1, E_2, E_3, \ldots \subseteq \mathbb{R}$,

$$\lambda^*\!\left(\bigcup_{k=1}^{\infty} E_k\right) \;\leq\; \sum_{k=1}^{\infty} \lambda^*(E_k). \tag{4}$$

Proof sketch. If any $\lambda^*(E_k) = +\infty$ the inequality is trivial. Otherwise, fix $\varepsilon > 0$. For each $k$, choose a countable cover of $E_k$ by open intervals with total length at most $\lambda^*(E_k) + \varepsilon/2^k$. The combined collection of all these intervals is a countable cover of $\bigcup_k E_k$, with total length at most

$$\sum_{k=1}^{\infty}\!\left(\lambda^*(E_k) + \frac{\varepsilon}{2^k}\right) = \sum_{k=1}^{\infty} \lambda^*(E_k) + \varepsilon.$$

Since $\varepsilon$ is arbitrary, $(4)$ follows.

These three properties — normalization $(2)$, monotonicity $(3)$, and countable sub-additivity $(4)$ — are the defining axioms of an outer measure in the abstract sense. Any function $\mu^* \colon 2^X \to [0, +\infty]$ satisfying them on a set $X$ is called an outer measure on $X$.

Outer measure agrees with length on intervals

A basic sanity check: for a bounded closed interval $[a, b]$,

$$\lambda^*([a, b]) = b - a. \tag{5}$$

Upper bound. Cover $[a, b]$ with the single interval $(a - \varepsilon, b + \varepsilon)$ to get $\lambda^*([a,b]) \leq b - a + 2\varepsilon$; take $\varepsilon \to 0$.

Lower bound. Any open cover of $[a, b]$ has a finite sub-cover (Heine–Borel). Summing the lengths of a finite cover of $[a, b]$ yields at least $b - a$ (a standard argument by induction on the number of overlapping intervals). Since every countable cover contains such a finite sub-cover, the infimum is at least $b - a$.

Together, $\lambda^*([a,b]) = b - a$. The same argument gives $\lambda^*((a,b)) = \lambda^*([a,b)) = b - a$ for open and half-open intervals.

Why countable additivity fails

Outer measure satisfies sub-additivity $(4)$, but it does not satisfy countable additivity on all subsets. The Vitali set $V$ from Introduction to Measure is the witness: the translates $V_q \coloneqq V + q$ (for $q \in \mathbb{Q} \cap [-1, 1]$) are pairwise disjoint, and

$$[0,1] \;\subseteq\; \bigcup_{q \in \mathbb{Q} \cap [-1,1]} V_q \;\subseteq\; [-1, 2].$$

Monotonicity gives $1 \leq \lambda^*\!\left(\bigcup_q V_q\right) \leq 3$. Since all $V_q$ are translates of $V$, they all have the same outer measure. If countable additivity held, the sum would be either $0$ (if $\lambda^*(V) = 0$) or $+\infty$ (if $\lambda^*(V) > 0$) — neither of which lies in $[1, 3]$. Contradiction.

The conclusion: outer measure is the best you can do on all subsets of $\mathbb{R}$. To obtain true additivity you must restrict to a suitable sub-collection — the measurable sets, identified by Carathéodory’s criterion.

Summary

• The Lebesgue outer measure $\lambda^*(E)$ is defined in equation $(1)$ as the infimum of total interval lengths over countable open covers of $E$.
• It satisfies: $\lambda^*(\emptyset) = 0$ (equation $(2)$), monotonicity (equation $(3)$), and countable sub-additivity (equation $(4)$).
• These three properties are the abstract definition of an outer measure; $\lambda^*$ is an outer measure on $\mathbb{R}$.
• On intervals, $\lambda^*$ agrees with ordinary length — equation $(5)$.
• Countable additivity fails on all subsets: the Vitali set shows that no consistent, translation-invariant, countably additive measure can live on $2^{\mathbb{R}}$.
• The fix is Carathéodory’s criterion: identify which sets split every test set additively, and restrict $\lambda^*$ to those sets.