Borel measure

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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 (some authors also require it to be tight). 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[edit]

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).

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