Borel measure

In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. Let X {displaystyle X} be a locally compact Hausdorff space, and let B ( X ) {displaystyle {mathfrak {B}}(X)} be the smallest σ-algebra that contains the open sets of X {displaystyle X} ; this is known as the σ-algebra of Borel sets. A Borel measure is any measure μ {displaystyle mu } defined on the σ-algebra of Borel sets. Some authors require in addition that μ {displaystyle mu } is locally compact, meaning that μ ( C ) < ∞ {displaystyle mu (C)<infty } for every compact set C {displaystyle C} . If a Borel measure μ {displaystyle mu } is both inner regular and outer regular, it is called a regular Borel measure (some authors also require it to be tight). If μ {displaystyle mu } is both inner regular and locally finite, it is called a Radon measure. The real line R {displaystyle mathbb {R} } with its usual topology is a locally compact Hausdorff space, hence we can define a Borel measure on it. In this case, B ( R ) {displaystyle {mathfrak {B}}(mathbb {R} )} is the smallest σ-algebra that contains the open intervals of R {displaystyle mathbb {R} } . While there are many Borel measures μ, the choice of Borel measure which assigns μ ( ( a , b ] ) = b − a {displaystyle mu ((a,b])=b-a} for every half-open interval ( a , b ] {displaystyle (a,b]} is sometimes called 'the' Borel measure on R {displaystyle mathbb {R} } . This measure turns out to be the restriction on the Borel σ-algebra of the Lebesgue measure λ {displaystyle lambda } , 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 which 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., λ ( E ) = μ ( E ) {displaystyle lambda (E)=mu (E)} for every Borel measurable set, where μ {displaystyle mu } is the Borel measure described above). If X and Y are second-countable, Hausdorff topological spaces, then the set of Borel subsets B ( X × Y ) {displaystyle B(X imes Y)} of their product coincides with the product of the sets B ( X ) × B ( Y ) {displaystyle B(X) imes B(Y)} of Borel subsets of X and Y. That is, the Borel functor from the category of second-countable Hausdorff spaces to the category of measurable spaces preserves finite products. The Lebesgue–Stieltjes integral is the ordinary Lebesgue integral with respect to a measure known as the Lebesgue–Stieltjes measure, which may be associated to any function of bounded variation on the real line. The Lebesgue–Stieltjes measure is a regular Borel measure, and conversely every regular Borel measure on the real line is of this kind. One can define the Laplace transform of a finite Borel measure μ on the real line by the Lebesgue integral An important special case is where μ is a probability measure or, even more specifically, the Dirac delta function. In operational calculus, the Laplace transform of a measure is often treated as though the measure came from a distribution function f. In that case, to avoid potential confusion, one often writes