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 $\mathbb {R}$ with its usual topology is a locally compact Hausdorff space, hence we can define a Borel measure on it. In this case, ${\mathfrak {B}}(\mathbb {R} )$ is the smallest σ-algebra that contains the open intervals of $\mathbb {R}$. 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 $\mathbb {R}$. 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.