# 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).^{[1]} Some authors require additional restrictions on the measure, as described below.

## Formal definition

Let *X* be a locally compact Hausdorff space, and let be the smallest σ-algebra that contains the open sets of *X*; this is known as the σ-algebra of Borel sets. A **Borel measure** is any measure *μ* defined on the σ-algebra of Borel sets.^{[2]} 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 means *μ*(*C*) < ∞ for every compact set *C*.

## On the real line

The real line with its usual topology is a locally compact Hausdorff space, hence we can define a Borel measure on it. In this case, is the smallest σ-algebra that contains the open intervals of . While there are many Borel measures μ, the choice of Borel measure which assigns for every half-open interval is sometimes called "the" Borel measure on . This measure turns out to be the restriction on the Borel σ-algebra of the Lebesgue measure , 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., for every Borel measurable set, where is the Borel measure described above).

## Product spaces

If *X* and *Y* are second-countable, Hausdorff topological spaces, then the set of Borel subsets of their product coincides with the product of the sets of Borel subsets of *X* and *Y*.^{[3]} That is, the Borel functor

from the category of second-countable Hausdorff spaces to the category of measurable spaces preserves finite products.

## Applications

### Lebesgue–Stieltjes integral

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.^{[4]}

### Laplace transform

One can define the Laplace transform of a finite Borel measure μ on the real line by the Lebesgue integral^{[5]}

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

where the lower limit of 0^{−} is shorthand notation for

This limit emphasizes that any point mass located at 0 is entirely captured by the Laplace transform. Although with the Lebesgue integral, it is not necessary to take such a limit, it does appear more naturally in connection with the Laplace–Stieltjes transform.

### Hausdorff dimension and Frostman's lemma

Given a Borel measure μ on a metric space *X* such that μ(*X*) > 0 and μ(*B*(*x*, *r*)) ≤ *r ^{s}* holds for some constant

*s*> 0 and for every ball

*B*(

*x*,

*r*) in

*X*, then the Hausdorff dimension dim

_{Haus}(

*X*) ≥

*s*. A partial converse is provided by Frostman's lemma:

^{[6]}

**Lemma:** Let *A* be a Borel subset of **R**^{n}, and let *s* > 0. Then the following are equivalent:

*H*^{s}(*A*) > 0, where*H*^{s}denotes the*s*-dimensional Hausdorff measure.- There is an (unsigned) Borel measure
*μ*satisfying*μ*(*A*) > 0, and such that

- holds for all
*x*∈**R**^{n}and*r*> 0.

### Cramér–Wold theorem

The Cramér–Wold theorem in measure theory states that a Borel probability measure on is uniquely determined by the totality of its one-dimensional projections.^{[7]} It is used as a method for proving joint convergence results. The theorem is named after Harald Cramér and Herman Ole Andreas Wold.

## References

- ↑ D. H. Fremlin, 2000.
*Measure Theory*. Torres Fremlin. - ↑ Alan J. Weir (1974).
*General integration and measure*. Cambridge University Press. pp. 158–184. ISBN 0-521-29715-X. - ↑ Vladimir I. Bogachev. Measure Theory, Volume 1. Springer Science & Business Media, Jan 15, 2007
- ↑ Halmos, Paul R. (1974),
*Measure Theory*, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90088-9 - ↑ Feller 1971, §XIII.1
- ↑ Rogers, C. A. (1998).
*Hausdorff measures*. Cambridge Mathematical Library (Third ed.). Cambridge: Cambridge University Press. pp. xxx+195. ISBN 0-521-62491-6. - ↑ K. Stromberg, 1994.
*Probability Theory for Analysts*. Chapman and Hall.

## Further reading

- Feller, William (1971),
*An introduction to probability theory and its applications. Vol. II.*, Second edition, New York: John Wiley & Sons, MR 0270403. - J. D. Pryce (1973).
*Basic methods of functional analysis*. Hutchinson University Library. Hutchinson. p. 217. ISBN 0-09-113411-0. - Ransford, Thomas (1995).
*Potential theory in the complex plane*. London Mathematical Society Student Texts.**28**. Cambridge: Cambridge University Press. pp. 209–218. ISBN 0-521-46654-7. Zbl 0828.31001. - Teschl, Gerald,
*Topics in Real and Functional Analysis*, (lecture notes)