Regular measure - Citizendia

In mathematics, a regular measure on a topological space is a measure for which every measurable set is "approximately open" and "approximately closed". Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with

1 Definition
2 Examples
3 References
4 See also

Definition

Let (X, T) be a topological space and let Σ be a σ-algebra on X that contains the topology T (so that all open and closed sets are measurable sets, and Σ is at least as fine as the Borel σ-algebra on X). In Mathematics, a σ-algebra (or sigma-algebra) ( Sigma is a Greek letter upper case Σ lower case σ over a set X is a nonempty In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in In Topology and related branches of Mathematics, a closed set is a set whose complement is open. In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with In Mathematics, the Borel algebra (or Borel &sigma-algebra) on a Topological space X is a &sigma-algebra of Subsets of Let μ be a measure on (X, Σ). A measurable subset A of X is said to be μ-regular if

$\mu (A) = \sup \{ \mu (F) | F \subseteq A, F \mbox{ closed} \}$

and

$\mu (A) = \inf \{ \mu (G) | G \supseteq A, F \mbox{ open} \}.$

Equivalently, A is a μ-regular set if and only if, for every δ > 0, there exists a closed set F and an open set G such that

$F \subseteq A \subseteq G$

and

$\mu (G \setminus F) < \delta.$

If every measurable set is regular, then the measure μ is said to be a regular measure. ↔

Examples

Lebesgue measure on the real line is a regular measure: see the regularity theorem for Lebesgue measure. In Mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a Length, Area or Volume to In Mathematics, the real line is simply the set R of singleton Real numbers However this term is usually used when R is to be treated as a In Mathematics, the regularity theorem for Lebesgue measure is a result in Measure theory that states that Lebesgue measure on the Real line
The trivial measure, which assigns measure zero to every measurable subset, is a regular measure. In Mathematics, specifically in Measure theory, the trivial measure on any Measurable space ( X, Σ is the measure μ which assigns
A trivial example of a non-regular measure is the measure μ on the real line with its usual Borel topology that assigns measure zero to the empty set and infinite positive measure to any non-empty set. In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members
Any Borel probability measure on any metric space is a regular measure. A probability space, in Probability theory, is the conventional Mathematical model of Randomness. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined

References

Billingsley, Patrick (1999). Convergence of Probability Measures. New York: John Wiley & Sons, Inc. . ISBN 0-471-19745-9.
Parthasarathy, K. R. (2005). Probability measures on metric spaces. AMS Chelsea Publishing, Providence, RI, pp. xii+276. ISBN 0-8218-3889-X. MR 2169627 (See chapter 2)

Contents

Definition

Examples

References

See also