Lebesgue Measure · aiwiki.page

Lebesgue measure is the standard way of assigning length, area, and volume to suitably measurable subsets of Euclidean space. On the line of real numbers, it assigns an interval its length; in higher dimensions, it assigns rectangular boxes the product of their side lengths. It is a central example in measure theory and supplies the underlying measure for the Lebesgue integral. Its construction extends elementary geometric measurement while retaining countable additivity and invariance under translations. The theory developed through Henri Lebesgue’s work on integration in 1902. (math.ucdavis.edu)

Construction and measurability

Write $m_n$ , or simply $m$ , for Lebesgue measure on $\mathbb R^n$ . Its construction begins with outer measure, which is defined for every subset, including those that will not be measurable. For an axis-aligned box

Q=\prod_{i=1}^{n}[a_i,b_i], \qquad |Q|=\prod_{i=1}^{n}(b_i-a_i),

define

m^*(E)= \inf\left\{ \sum_{k=1}^{\infty}|Q_k|: E\subseteq\bigcup_{k=1}^{\infty}Q_k \right\}.

The infimum ranges over countable box coverings. Covering boxes may overlap, and the resulting value may be infinite. Thus outer measure describes the least total covering volume attainable, rather than a particular covering. (math.ucdavis.edu)

A subset $E$ is Lebesgue measurable when it satisfies the Carathéodory criterion:

m^*(A)=m^*(A\cap E)+m^*(A\setminus E) \quad\text{for every }A\subseteq\mathbb R^n.

This condition says that splitting any test set along $E$ preserves its outer measure additively. The measurable sets form a sigma-algebra, denoted $\mathcal L^n$ , and Lebesgue measure is the restriction $m=m^*|_{\mathcal L^n}$ . Outer measure is countably subadditive on arbitrary sets; its restriction is countably additive on measurable sets. (math.ucdavis.edu)

Additivity and geometric invariance

For pairwise disjoint measurable sets $E_1,E_2,\ldots$ ,

m\left(\bigcup_{k=1}^{\infty}E_k\right) =\sum_{k=1}^{\infty}m(E_k).

Also, $m(\varnothing)=0$ , and $E\subseteq F$ implies $m(E)\leq m(F)$ . The measure is sigma-finite: although $m(\mathbb R^n)=\infty$ , Euclidean space is a countable union of bounded boxes of finite measure. (math.ucdavis.edu)

Lebesgue measure is translation invariant:

m(E+x)=m(E).

Rotations and reflections also preserve it. Dilation by a nonzero real scalar $r$ gives

m(rE)=|r|^n m(E).

More generally, an invertible linear map $T:\mathbb R^n\to\mathbb R^n$ satisfies

m(T(E))=|\det T|\,m(E).

The absolute value of its determinant is therefore the volume-scaling factor. These properties connect the abstract construction with ordinary geometric measurement. (ocw.mit.edu)

Null sets, completeness, and regularity

A null set has measure zero. Every countable set is null: its points can be covered by boxes whose total volume is arbitrarily small. Consequently, the rational numbers have measure zero despite forming a dense set in the real line. Uncountability does not imply positive measure; the standard Cantor set is uncountable and null. (math.ucdavis.edu)

Lebesgue measure is complete, meaning that every subset of a null set is measurable and null. All sets in the Borel sigma-algebra, generated by open sets, are Lebesgue measurable, but the converse fails. Indeed, Lebesgue measure is the completion of its restriction to Borel sets: each Lebesgue measurable set differs from a Borel set by a null set. (math.ucdavis.edu)

The measure is also regular. For every measurable $E$ ,

m(E)=\inf_{\substack{E\subseteq U\\U\text{ open}}}m(U) =\sup_{\substack{K\subseteq E\\K\text{ compact}}}m(K).

Thus measurable sets can be approximated externally by open sets and internally by compact sets. For a finite-measure set and any $\varepsilon>0$ , these approximations can be chosen with $m(U\setminus K)<\varepsilon$ . (math.ucdavis.edu)

Nonmeasurable sets

Not every subset of $\mathbb R$ is Lebesgue measurable. A Vitali set is constructed using the equivalence relation $x\sim y$ when $x-y$ is rational. The axiom of choice permits selecting one representative from each equivalence class meeting $[0,1]$ . (ma.imperial.ac.uk)

Its translates by rational numbers in $[-1,1]$ are pairwise disjoint, cover $[0,1]$ , and lie within $[-1,2]$ . If the selected set were measurable, translation invariance would give every translate the same measure. Countable additivity would then make their union have either zero or infinite measure, contradicting those containments. This shows why Lebesgue measure cannot be defined on every subset while preserving both interval lengths and its defining additivity and invariance properties. (ma.imperial.ac.uk)

Integration and probability

Lebesgue measure supplies the set sizes used to integrate measurable functions. For a nonnegative simple function $s=\sum_j a_j\mathbf1_{E_j}$ , with disjoint measurable $E_j$ ,

\int s\,dm=\sum_j a_jm(E_j).

General nonnegative integrals are obtained through approximation by simple functions; signed functions are treated through their positive and negative parts. A property holds almost everywhere when its exceptional set is null. Changing a function on such a set does not change its Lebesgue integral. (ocw.mit.edu)

In probability, a probability density function $f$ on $\mathbb R^n$ defines probabilities through

P(A)=\int_A f\,dm, \qquad f\geq0,\qquad \int_{\mathbb R^n}f\,dm=1.

Lebesgue measure itself is not a probability measure on the whole space, since its total mass is infinite. Restricted to $[0,1]$ , however, it has total mass one and yields the uniform distribution. For distributions with a density, individual points have probability zero even though intervals may have positive probability. (ocw.mit.edu)