Borel measure
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-algbera 
(the Borel σ-algebra on E) is Borel iff 
compact.