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.

Formal definition

Let $X$ be a locally compact Hausdorff space, and let $\mathfrak(X)$ 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 $\mu$ defined on the σ-algebra of Borel sets. A few authors require in addition that $\mu$ is locally finite, meaning that $\mu(C)<\infty$ for every compact set $C$. If a Borel measure $\mu$ is both inner regular and outer regular, it is called a regular Borel measure. If $\mu$ is both inner regular, outer regular, and locally finite, it is called a Radon measure.

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, $\mathfrak(\mathbb R)$ 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 that assigns $\mu((a,b])=b-a$ for every half-open interval $(a,b]$ is sometimes called "the" Borel measure on $\mathbb R$. This measure turns out to be the restriction to the Borel σ-algebra of the Lebesgue measure $\lambda$, 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 that 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., $\lambda(E)=\mu(E)$ for every Borel measurable set, where $\mu$ is the Borel measure described above).

Product spaces

If ''X'' and ''Y'' are second-countable, Hausdorff topological spaces, then the set of Borel subsets $B(X\times Y)$ of their product coincides with the product of the sets $B(X)\times B(Y)$ of Borel subsets of ''X'' and ''Y''. That is, the Borel functor
: $\mathbf\colon\mathbf_\to\mathbf$
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.

Laplace transform

One can define the Laplace transform of a finite Borel measure μ on the real line by the Lebesgue integral

: $(\mathcal\mu)(s) = \int_ e^\,d\mu(t).$

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

: $(\mathcalf)(s) = \int_^\infty e^f(t)\,dt$

where the lower limit of 0⁻ is shorthand notation for

: $\lim_\int_^\infty.$

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 the Frostman lemma:

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
::$\mu(B(x,r))\le r^s$
: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 $\mathbb R^k$ is uniquely determined by the totality of its one-dimensional projections.K. Stromberg, 1994. ''Probability Theory for Analysts''. Chapman and Hall. It is used as a method for proving joint convergence results. The theorem is named after Harald Cramér and Herman Ole Andreas Wold.

References

Further reading

* Gaussian measure, a finite-dimensional Borel measure
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* Wiener's lemma related

External links

Borel measure
a
Encyclopedia of Mathematics

{{DEFAULTSORT:Borel Measure
Measures (measure theory)