Топологиялық кеңістіктегі өлшемдер туралы

Measure defined on all open sets of a topological space
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 be a locally compact Hausdorff space, and let be the smallest σ algebra that contains the open sets of ; this is known as the σ algebra of Borel sets. A Borel measure is any measure defined on the σ algebra of Borel sets. A few authors require in addition that is locally finite, meaning that for every compact set If a Borel measure is both inner regular and outer regular, it is called a regular Borel measure. If is both inner regular, outer regular, and locally finite, it is called a Radon measure.

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 that assigns for every half open interval is sometimes called "the" Borel measure on This measure turns out to be the restriction to 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 that contains all the Borel sets and can be equipped with a complete measure. 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). This idea extends to finite dimensional spaces (the Cramér–Wold theorem, below) but does not hold, in general, for infinite dimensional spaces. Infinite dimensional Lebesgue measures do not exist.

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. 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.

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. It is used as a method for proving joint convergence results. The theorem is named after Harald Cramér and Herman Ole Andreas Wold.