- Borel measure
mathematics, the Borel algebrais 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 measureis 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 regularand outer regularon all Borel sets, it is called a regular Borel measure.
If μ is inner regular and locally finite, μ is said to be a
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