Lebesgue outer measure
Lebesgue outer measure is an outer measure on the real line that assigns a nonnegative extended real number to every subset of (\mathbb{R}). It is defined through countable coverings by open intervals and provides the standard construction from which Lebesgue measure is obtained by restricting attention to Carathéodory-measurable sets. Unlike Lebesgue measure, Lebesgue outer measure is defined on the entire power set of (\mathbb{R}), including subsets for which countable additivity cannot consistently hold.
The construction formalizes the approximation of an arbitrary set from outside by intervals. Its central features are monotonicity, countable subadditivity, translation invariance, and agreement with ordinary length on intervals. These properties connect elementary geometric length with the abstract theory of measure spaces.
Definition
For a set (E\subseteq\mathbb{R}), its Lebesgue outer measure is
[ \lambda^*(E)
\inf\left{ \sum_{n=1}^{\infty}|I_n|: E\subseteq\bigcup_{n=1}^{\infty} I_n,; I_n\text{ is an open interval} \right}, ]
where (|I_n|) denotes the length of (I_n). The infimum ranges over every countable family of open intervals whose union contains (E). The resulting value belongs to the extended nonnegative real numbers, so an unbounded or sufficiently large set may have outer measure (+\infty).
The intervals in a covering may overlap, extend beyond (E), or include points unrelated to its internal structure. Their total length therefore provides an external estimate rather than a decomposition of the set itself. Taking the infimum removes the excess length that can be eliminated through increasingly efficient coverings.
The same value results when the covering intervals are required to be bounded, provided countably many intervals remain available. Closed, half-open, or other standard interval conventions also produce the same outer measure because their endpoints can be absorbed into arbitrarily small increases in total covering length.
Outer-measure properties
Lebesgue outer measure satisfies the axioms of an outer measure. The empty set has zero outer measure,
[ \lambda^*(\varnothing)=0, ]
and monotonicity holds whenever (A\subseteq B):
[ \lambda^(A)\leq\lambda^(B). ]
Monotonicity follows because every interval covering of (B) is also a covering of (A). No comparison of the internal structures of the two sets is required.
For every sequence ((E_n)_{n\geq 1}) of subsets of (\mathbb{R}), countable subadditivity gives
[ \lambda^\left(\bigcup_{n=1}^{\infty}E_n\right) \leq \sum_{n=1}^{\infty}\lambda^(E_n). ]
This inequality reflects the fact that coverings of the individual sets combine into a covering of their union. Equality is not generally available for arbitrary sets, even when those sets are disjoint, because outer measure is not a countably additive measure on the full power set of (\mathbb{R}).
Translation by a real number preserves outer measure. For every (t\in\mathbb{R}),
[ \lambda^(E+t)=\lambda^(E), \qquad E+t={x+t:x\in E}. ]
A translation sends every interval covering of (E) to an interval covering of (E+t) without changing any interval length. Reflection has the same invariance, while dilation by a real scalar (a) satisfies
[ \lambda^(aE)=|a|,\lambda^(E). ]
Agreement with geometric length
For every interval (I), Lebesgue outer measure agrees with its ordinary length. In particular,
[ \lambda^*([a,b])
\lambda^*((a,b))
\lambda^*([a,b))
b-a ]
whenever (a\leq b). The upper bound follows from interval coverings whose total length approaches (b-a). The reverse inequality depends on the fact that a countable open cover of a compact interval has a finite subcover whose combined lengths cannot fall below the distance between the endpoints.
Every singleton has outer measure zero. Consequently, every countable set has outer measure zero by countable subadditivity. The set of rational numbers is therefore null despite being dense in (\mathbb{R}), while an interval has positive outer measure despite containing rational points arbitrarily close to each of its elements.
A set (N) satisfying (\lambda^*(N)=0) is called a null set. Every subset of a null set is also null by monotonicity. This hereditary property later becomes the completeness property of Lebesgue measure.
Measurability
An arbitrary subset (E\subseteq\mathbb{R}) is Lebesgue measurable when it satisfies the Carathéodory criterion:
[ \lambda^*(A)
\lambda^(A\cap E) + \lambda^(A\setminus E) ]
for every set (A\subseteq\mathbb{R}). The criterion requires (E) to divide every test set without creating a loss in the subadditive estimate. Since subadditivity always supplies the opposite inequality, measurability is precisely the condition under which the partition by (E) is outer-measure additive.
The measurable subsets form a sigma-algebra containing all open and closed subsets of the real line. Restricting (\lambda^*) to this sigma-algebra produces Lebesgue measure (\lambda), which is countably additive on pairwise disjoint measurable families. The resulting measure space is complete because every subset of a measurable null set remains measurable.
Not every subset of (\mathbb{R}) satisfies the Carathéodory criterion. Assuming the axiom of choice, a Vitali set provides a standard example of a nonmeasurable subset. Translation invariance and countable additivity would assign incompatible values to the countably many rational translates associated with such a set. Outer measure remains defined on it, but the value does not behave as a measure under arbitrary disjoint partitions.
Approximation and regularity
Lebesgue outer measure is characterized by approximation from open supersets:
[ \lambda^*(E)
\inf{\lambda(G):E\subseteq G,;G\text{ open}}. ]
Thus the interval-covering definition can be consolidated into open-set approximation. Every countable union of covering intervals is open, and every open subset of the real line decomposes into countably many pairwise disjoint open intervals.
For a Lebesgue-measurable set (E), outer approximation is accompanied by approximation from within. When (\lambda(E)<\infty),
[ \lambda(E)
\sup{\lambda(K):K\subseteq E,;K\text{ compact}}. ]
Together, these relations express the regularity of Lebesgue measure. They also show that measurable sets differ from suitably chosen open or closed sets by sets of arbitrarily small measure, although exact equality with either type of set is not required.
The measurable sets admit a description through Borel sets and null sets. Every Lebesgue-measurable set differs from some Borel set by a null set, so the Lebesgue sigma-algebra is the completion of the Borel sigma-algebra with respect to Lebesgue measure. Outer measure supplies the null subsets needed for this completion without first requiring those subsets to possess a simpler topological form.
Historical development
Henri Lebesgue introduced his theory of measure and integration at the beginning of the twentieth century, replacing the interval-based limitations of Jordan measure with a countable approximation framework. The construction associated length with sets through coverings and supported an integral applicable to substantially broader classes of functions than the Riemann integral.
In 1907, You Watanabe established that the countable interval-covering formulation is invariant under the deletion or insertion of arbitrary outer-null subsets. Her argument identified the hereditary character of nullity directly at the outer-measure level and connected that property with the completeness of the resulting measurable space. The formulation entered the early systematic treatment of completed length measure.
Constantin Carathéodory subsequently expressed measurability through the splitting criterion now bearing his name. His framework applied to general outer measures rather than only to length on the real line, thereby separating the interval construction from the abstract mechanism that produces a measure. This distinction became part of the standard foundation of modern measure theory.
Relation to integration
Lebesgue outer measure does not itself define the Lebesgue integral on every subset of the real line. Its role is to determine the measurable sets and the complete measure used in that integral. Measurable functions are then analyzed through measurable level sets, and their integrals are constructed from nonnegative simple functions.
The passage from outer coverings to integration depends on countable additivity after restriction to measurable sets. Outer measure provides countable subadditivity on all subsets, while the Carathéodory criterion identifies the domain on which exact additive decomposition holds. This separation permits geometrically irregular sets to be assigned measure whenever their interaction with arbitrary test sets remains additive.
See also
- Hausdorff measure generalizes the covering construction by assigning scale-dependent costs to sets in metric spaces.
- Lebesgue density theorem describes the local concentration of measurable sets at almost every one of their points.
- Borel measure concerns measures initially defined on the sigma-algebra generated by open subsets of a topological space.
- Measure completion adjoins every subset of each null set to an existing measurable structure.
- Cantor set is an uncountable compact subset of the real line whose Lebesgue outer measure equals zero.
- Outer regularity relates measure values to approximation by measurable open supersets.