# 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. Any measure *μ* defined on the σ-algebra of Borel sets is called a **Borel measure**.^{[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 automatically satisfies *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 interval is sometimes called "the" Borel measure on . In practice, even "the" Borel measure is not the most useful measure defined on the σ-algebra of Borel sets; indeed, the Lebesgue measure 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 is an extension of the Borel measure , 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., for every Borel measurable set).

## Applications

### Lebesgue-Stieltjes integral

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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.^{[3]}

### Laplace transform

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One can define the Laplace transform of a finite Borel measure μ on the real line by the Lebesgue integral^{[4]}

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

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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:

^{[5]}

**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

### Cramér–Wold theorem

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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.^{[6]} 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. - ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}
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- ↑ K. Stromberg, 1994.
*Probability Theory for Analysts*. Chapman and Hall.

## Further reading

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