Borel measure

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 &sigma;-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 &mu; on the &sigma;-algebra mathfrak{B}(X) of Borel sets &mdash; the Borel &sigma;-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.



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