Editing Borel measure (section)

==On the real line==
The [[real line]] <math>\mathbb R</math> with its [[Real line#As a topological space|usual topology]] is a locally compact Hausdorff space; hence we can define a Borel measure on it. In this case, <math>\mathfrak{B}(\mathbb R)</math> is the smallest σ-algebra that contains the [[open interval]]s of <math>\mathbb R</math>. While there are many Borel measures ''μ'', the choice of Borel measure that assigns <math>\mu((a,b])=b-a</math> for every half-open interval <math>(a,b]</math> is sometimes called "the" Borel measure on <math>\mathbb R</math>. This measure turns out to be the restriction to the Borel σ-algebra of the [[Lebesgue measure]] <math>\lambda</math>, which is a [[complete measure]] and is defined on the Lebesgue σ-algebra. The Lebesgue σ-algebra is actually the ''completion'' of the Borel σ-algebra, which means that it is the smallest σ-algebra that contains all the Borel sets and can be equipped with a [[complete measure]]. Also, the Borel measure and the Lebesgue measure coincide on the Borel sets (i.e., <math>\lambda(E)=\mu(E)</math> for every Borel measurable set, where <math>\mu</math> is the Borel measure described above). This idea extends to finite-dimensional spaces <math>\mathbb R^n</math> (the [[Cramér–Wold theorem]], below) but does not hold, in general, for infinite-dimensional spaces. [[Infinite-dimensional Lebesgue measure]]s do not exist.