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 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 context, Let X be a locally compact Hausdorff space. A Borel measure is any measure μ on the σ-algebra $\backslash mathfrak\{B\}(X)$, the Borel σ-algebra on X.

If μ is both inner regular and outer regular on all Borel sets, it is called a regular Borel measure.

If μ is outer regular on Borel sets, inner regular on open sets, and all compact Borel sets have finite measure, μ is said to be a Radon measure.

Measure theory

Borel-Maß | Vikipedio:Projekto matematiko/Borela mezuro | Mesure de Borel | Misura di Borel | Miara borelowska | 波莱尔测度