Borel measure

From Free net encyclopedia

In mathematics, the Borel algebra is the smallest σ-algebra on the real numbers R containing the intervals, and the Borel measure is the measure on this σ-algebra which gives to the interval [a, b] the measure b − a (where a < b).

The Borel measure is not complete, which is why in practice the complete Lebesgue measure is preferred: every Borel measurable set is also Lebesgue measurable, and the measures of the set agree.

In a more general (abstract) measure-theoretic context, Let E be a Hausdorff space. A measure μ on the σ-algebra <math>\mathfrak{B}(E) </math> (the Borel σ-algebra on E) is Borel iff <math>\mu(K) < +\infty</math> for all <math>K \subset E</math> compact.

See also: Borel setde:Borel-Maß eo:Vikipedio:Projekto matematiko/Borela mezuro fr:Mesure de Borel it:Misura di Borel zh:波莱尔測度