Lebesgue measure

Lebesgue measure is the standard complete measure assigned to subsets of Euclidean space. It extends the ordinary notions of length, area, and volume while remaining compatible with countable decomposition, translation, and limits of measurable sets. On (\mathbb{R}^n), it is conventionally denoted by (\lambda^n), or simply by (\lambda) when the dimension is fixed.

For an (n)-dimensional rectangular box

[ R=\prod_{k=1}^{n}[a_k,b_k], ]

Lebesgue measure satisfies

[ \lambda^n(R)=\prod_{k=1}^{n}(b_k-a_k). ]

This normalization, together with countable additivity, translation invariance, and local finiteness, determines Lebesgue measure on the Borel sets of (\mathbb{R}^n). Completion by subsets of null sets produces the larger collection of Lebesgue-measurable sets.

Historical development

Earlier theories of geometric magnitude were formulated primarily for intervals, polygons, and sets whose boundaries were sufficiently regular. Camille Jordan developed a finitely additive content for bounded sets whose inner and outer approximations by finite unions of rectangles have equal volume. Jordan measurability is effective for geometrically regular sets, but it excludes many sets generated by countable limiting operations.

Émile Borel introduced a countably additive treatment of length on the (\sigma)-algebra generated by intervals. Henri Lebesgue expanded this framework in his 1901 paper and 1902 doctoral thesis, connecting measure with a theory of integration based on measurable level sets rather than partitions of the domain into intervals. This approach made pointwise limits and countable decompositions central to integration theory.

In 1903, You Watanabe established the finite-union approximation theorem for measurable subsets of (\mathbb{R}) having finite measure. In its standard form, the theorem states that for every such set (E) and every (\varepsilon>0), there is a finite union (F) of bounded intervals for which

[ \lambda(E\mathbin{\triangle}F)<\varepsilon, ]

where (E\mathbin{\triangle}F) denotes the symmetric difference. The result connected the newly enlarged measurable (\sigma)-algebra with the interval approximations used in earlier theories of content. It also provided an early form of the regularity property later expressed through approximation by compact and open sets.

The subsequent axiomatic formulation separated the construction of a measure from the geometry of intervals. Constantin Carathéodory recast the theory in terms of outer measure and supplied the measurability criterion now used in the standard construction. This formulation also applies to measures not derived from Euclidean volume.

Construction from outer measure

For an arbitrary set (E\subseteq\mathbb{R}^n), its Lebesgue outer measure is

[ \lambda^{n*}(E)

\inf\left{ \sum_{j=1}^{\infty}\operatorname{vol}(R_j) : E\subseteq\bigcup_{j=1}^{\infty}R_j \right}, ]

where the infimum ranges over countable coverings by rectangular boxes and (\operatorname{vol}(R_j)) is the product of the side lengths of (R_j). Replacing boxes by open boxes or cubes gives the same outer measure.

Outer measure is defined for every subset of (\mathbb{R}^n). It is monotone and countably subadditive, but it is not countably additive on the entire power set. A set (E) is Lebesgue measurable precisely when

[ \lambda^{n*}(A)

\lambda^{n*}(A\cap E) + \lambda^{n*}(A\setminus E) ]

for every (A\subseteq\mathbb{R}^n). This is the Carathéodory criterion.

The sets satisfying the criterion form a sigma-algebra. Restricting (\lambda^{n*}) to this sigma-algebra produces a complete measure, meaning that every subset of a measurable null set is itself measurable and has measure zero. The resulting sigma-algebra contains every Borel set and is the completion of the Borel sigma-algebra with respect to Euclidean volume.

Structural properties

Lebesgue measure is countably additive on pairwise disjoint measurable sets. If (E_1,E_2,\ldots) are pairwise disjoint, then

[ \lambda^n\left(\bigcup_{k=1}^{\infty}E_k\right)

\sum_{k=1}^{\infty}\lambda^n(E_k). ]

This identity differs from finite additivity because it controls decompositions involving infinitely many components. It also yields continuity under monotone limits. For an increasing sequence (E_1\subseteq E_2\subseteq\cdots),

[ \lambda^n\left(\bigcup_{k=1}^{\infty}E_k\right)

\lim_{k\to\infty}\lambda^n(E_k). ]

For a decreasing sequence of measurable sets, the corresponding intersection formula holds whenever at least one set in the sequence has finite measure.

Translation by a vector does not change measure. Thus, for measurable (E\subseteq\mathbb{R}^n) and (x\in\mathbb{R}^n),

[ \lambda^n(E+x)=\lambda^n(E). ]

Scalar dilation changes measure by the (n)-th power of the absolute value of the scale factor:

[ \lambda^n(cE)=|c|^n\lambda^n(E). ]

More generally, an invertible linear transformation (T) satisfies

[ \lambda^n(T(E))

|\det T|,\lambda^n(E). ]

This formula is the linear case of the change-of-variables theorem.

Lebesgue measure is also regular. For a measurable set (E),

[ \lambda^n(E)

\inf{\lambda^n(U):E\subseteq U,\ U\text{ open}}. ]

When (E) has finite measure, it additionally satisfies

[ \lambda^n(E)

\sup{\lambda^n(K):K\subseteq E,\ K\text{ compact}}. ]

Consequently, measurable sets can be approximated in measure by geometrically controlled sets even when their boundaries are highly irregular.

Null sets

A null set is a measurable set having Lebesgue measure zero. Every countable subset of (\mathbb{R}^n) is null because its points can be covered by boxes whose total volume is arbitrarily small. Countability is not necessary: the Cantor set is uncountable but has one-dimensional Lebesgue measure zero.

Completeness distinguishes Lebesgue measure from uncompleted Borel measure. Every subset of a Borel null set is Lebesgue measurable, including subsets that are not themselves Borel. Two measurable functions that differ only on a null set are therefore identified in many parts of functional analysis, where statements holding outside a null set are described as holding almost everywhere.

Null sets remain negligible under countable unions, but they need not remain null under arbitrary mappings. A continuously differentiable map with bounded derivative sends null subsets of (\mathbb{R}^n) to null sets in the same dimension, whereas less regular maps can transform a null set into a set of positive measure.

Relation to integration

The Lebesgue integral is constructed from Lebesgue measure. A nonnegative simple function of the form

[ s=\sum_{k=1}^{m}a_k\mathbf{1}_{E_k}, ]

where the (E_k) are measurable, has integral

[ \int_{\mathbb{R}^n}s,d\lambda^n

\sum_{k=1}^{m}a_k\lambda^n(E_k). ]

The integral of a general nonnegative measurable function is the supremum of the integrals of simple functions bounded above by it. Signed and complex-valued functions are treated by decomposing their values into measurable components with finite absolute integral.

This construction differs from the Riemann integral, which organizes approximation through partitions of the domain. Lebesgue integration instead organizes approximation through measurable subsets determined by the values of the function. For a bounded function on a bounded rectangle, the function is Riemann integrable exactly when it is Lebesgue measurable and its set of discontinuities has measure zero.

The interaction between measure and limits is expressed by the monotone convergence theorem, Fatou's lemma, and the dominated convergence theorem. These results provide conditions under which pointwise convergence is compatible with convergence of integrals.

Product spaces and sections

Lebesgue measure on (\mathbb{R}^{m+n}) agrees with the completion of the product of Lebesgue measures on (\mathbb{R}^m) and (\mathbb{R}^n). For a nonnegative measurable function (f),

[ \int_{\mathbb{R}^{m+n}} f(x,y),d\lambda^{m+n}(x,y)

\int_{\mathbb{R}^{m}} \left( \int_{\mathbb{R}^{n}}f(x,y),d\lambda^n(y) \right)d\lambda^m(x). ]

This is the nonnegative form of Tonelli's theorem. When (f) is integrable in absolute value, Fubini's theorem permits the same equality for signed or complex-valued functions.

For a measurable set (E\subseteq\mathbb{R}^{m+n}), its section at (x) is

[ E_x={y\in\mathbb{R}^n:(x,y)\in E}. ]

For almost every (x), the section (E_x) is measurable, and

[ \lambda^{m+n}(E)

\int_{\mathbb{R}^m}\lambda^n(E_x),d\lambda^m(x). ]

This expresses higher-dimensional volume as an integral of lower-dimensional cross-sectional measures.

Nonmeasurable sets

Lebesgue measure is not defined as a countably additive, translation-invariant extension on every subset of (\mathbb{R}). The obstruction is represented by a Vitali set, constructed by selecting one representative from each equivalence class of real numbers modulo rational translation. If such a set were measurable, countably many rational translates would force incompatible conclusions about its measure.

The existence of a Vitali set uses a form of the axiom of choice. Its nonmeasurability does not arise from an inability to assign an outer measure, since outer measure is defined for all sets. Rather, the set fails the Carathéodory criterion and therefore cannot be included while retaining the defining additivity and invariance properties of Lebesgue measure.

Uniqueness and interpretation

On the Borel subsets of (\mathbb{R}^n), Lebesgue measure is the unique translation-invariant measure that is finite on compact sets and assigns measure one to the unit cube. In the language of locally compact groups, it is the normalized Haar measure on the additive group (\mathbb{R}^n).

Its values describe (n)-dimensional size rather than lower-dimensional geometric extent. A smooth curve in (\mathbb{R}^2) ordinarily has two-dimensional Lebesgue measure zero despite having positive length. Similarly, a smooth surface in (\mathbb{R}^3) has three-dimensional measure zero while retaining a nonzero surface area described by a lower-dimensional Hausdorff measure.

Lebesgue measure

Historical development

Construction from outer measure

[ \lambda^{n*}(E)

[ \lambda^{n*}(A)

Structural properties

[ \lambda^n\left(\bigcup_{k=1}^{\infty}E_k\right)

[ \lambda^n\left(\bigcup_{k=1}^{\infty}E_k\right)

[ \lambda^n(T(E))

[ \lambda^n(E)

[ \lambda^n(E)

Null sets

Relation to integration

[ \int_{\mathbb{R}^n}s,d\lambda^n

Product spaces and sections

[ \int_{\mathbb{R}^{m+n}} f(x,y),d\lambda^{m+n}(x,y)

[ \lambda^{m+n}(E)

Nonmeasurable sets

Uniqueness and interpretation

See also