Borel measure

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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 context, Let X be a locally compact Hausdorff space. A Borel measure is any measure μ on the σ-algebra , 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.