Borel measure

In mathematics, specifically in measure theory, a Borel measure is defined as follows: let X be a locally compact Hausdorff space, and let \( \mathfrak{B}(X) \) be the smallest σ-algebra that contains the open sets of X; this is known as the σ-algebra of Borel sets. Any measure μ defined on the σ-algebra of Borel sets is called a Borel measure. Some authors require in addition that μ(C) < ∞ for every compact set C. If a Borel measure μ is both inner regular and outer regular, it is called a regular Borel measure. If μ is both inner regular and locally finite, it is called a Radon measure. Note that a locally finite Borel measure automatically satisfies μ(C) < ∞ for every compact set C.
On the real line

The real line R with its usual topology is a locally compact Hausdorff space, hence we can define a Borel measure on it. In this case, \( \mathfrak{B}(\textbf{R}) \) is the smallest σ-algebra that contains the open intervals of R. While there are many Borel measures μ, the choice of Borel measure which assigns \( \mu([a,b])=b-a \) for every interval [a,b] is sometimes called "the" Borel measure on R. In practice, even "the" Borel measure is not the most useful measure defined on the σ-algebra of Borel sets; indeed, the Lebesgue measure\( \lambda \) is an extension of "the" Borel measure which possesses the crucial property that it is a complete measure (unlike the Borel measure). To clarify, when one says that the Lebesgue measure \( \lambda \) is an extension of the Borel measure \( \mu \), it means that every Borel measurable set E is also a Lebesgue measurable set, and the Borel measure and the Lebesgue measure coincide on the Borel sets (i.e., \( \lambda(E)=\mu(E) \)for every Borel measurable set).
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.

Mathematics Encyclopedia