Inner regular measure
Encyclopedia
In mathematics
, an inner regular measure is one for which the measure
of a set can be approximated from within by compact
subset
s.
topological space
and let Σ be a σ-algebra on X that contains the topology T (so that every open set
is a measurable set, and Σ is at least as fine as the Borel σ-algebra on X). Then a measure μ on the measurable space (X, Σ) is called inner regular if, for every set A in Σ,
This property is sometimes referred to in words as "approximation from within by compact sets."
Some authors use the term tight as a synonym
for inner regular. This use of the term is closely related to tightness of a family of measures
, since a measure μ is inner regular if and only if
, for all ε > 0, there is some compact subset
K of X such that μ(X \ K) < ε. This is precisely the condition that the singleton collection of measures {μ} is tight.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an inner regular measure is one for which the measure
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
of a set can be approximated from within by compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
s.
Definition
Let (X, T) be a HausdorffHausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
and let Σ be a σ-algebra on X that contains the topology T (so that every open set
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
is a measurable set, and Σ is at least as fine as the Borel σ-algebra on X). Then a measure μ on the measurable space (X, Σ) is called inner regular if, for every set A in Σ,
This property is sometimes referred to in words as "approximation from within by compact sets."
Some authors use the term tight as a synonym
Synonym
Synonyms are different words with almost identical or similar meanings. Words that are synonyms are said to be synonymous, and the state of being a synonym is called synonymy. The word comes from Ancient Greek syn and onoma . The words car and automobile are synonyms...
for inner regular. This use of the term is closely related to tightness of a family of measures
Tightness of measures
In mathematics, tightness is a concept in measure theory. The intuitive idea is that a given collection of measures does not "escape to infinity."-Definitions:...
, since a measure μ is inner regular if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
, for all ε > 0, there is some compact subset
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
K of X such that μ(X \ K) < ε. This is precisely the condition that the singleton collection of measures {μ} is tight.