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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 $mathfrak\{B\}(X)$, 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'.