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.

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 of Borel sets — 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 inner regular and locally finite, μ is said to be a Radon measure.

References

• J.D. Pryce (1973). Basic methods of functional analysis. Hutchinson University Library. Hutchinson. p. 217. ISBN 0-09-113411-0. 
• Alan J. Weir (1974). General integration and measure. Cambridge University Press. pp. 158–184. ISBN 0-521-29715-X.