Borel measure
The real line formula_10 with its usual topology is a locally compact Hausdorff space, hence we can define a Borel measure on it. In this case, formula_11 is the smallest σ-algebra that contains the open intervals of formula_10. While there are many Borel measures "μ", the choice of Borel measure that assigns formula_13 for every half-open interval formula_14 is sometimes called "the" Borel measure on formula_10. This measure turns out to be the restriction to the Borel σ-algebra of the Lebesgue measure formula_16, which is a complete measure and is defined on the Lebesgue σ-algebra. The Lebesgue σ-algebra is actually the "completion" of the Borel σ-algebra, which means that it is the smallest σ-algebra that contains all the Borel sets and has a complete measure on it. Also, the Borel measure and the Lebesgue measure coincide on the Borel sets (i.e., formula_17 for every Borel measurable set, where formula_4 is the Borel measure described above).