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In mathematics, an inner regular measure is one for which the measure of a set can be approximated from within by compact subsets. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
In mathematics, a measure is a function that assigns a number, e. ...
In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ...
A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ...
Definition Let (X, T) be a Hausdorff 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, In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...
Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
In mathematics, a σ-algebra (pronounced sigma-algebra) or σ-field over a set X is a collection Σ of subsets of X that is closed under countable set operations; σ-algebras are mainly used in order to define measures on X. The concept is important in mathematical analysis and probability theory. ...
In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...
In mathematics, a measure is a function that assigns a number, e. ...
In mathematics, the Borel algebra (or Borel σ-algebra) on a topological space is either of two σ-algebras on a topological space X: The minimal σ-algebra containing the open sets. ...
In mathematics, a σ-algebra (or σ-field) X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S. The concept is important in mathematical analysis and probability theory. ...
This property is sometimes referred to in words as "approximation from within by compact sets."
Some authors[1] 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. Synonyms can be nouns, adverbs or adjectives, as long as both members of the pair are the same part of speech. ...
In mathematics, tightness is a concept in measure theory, the intuitive idea being that a given collection of measures does not escape to infinity. ...
It has been suggested that this article or section be merged with Logical biconditional. ...
In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ...
In mathematics, a singleton is a set with exactly one element. ...
Reference - ^ Ambrosio, L., Gigli, N. & Savaré, G. (2005). Gradient Flows in Metric Spaces and in the Space of Probability Measures. Basel: ETH Zürich, Birkhäuser Verlag. ISBN 3-7643-2428-7.
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