In the mathematical field of measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the...

subset E ⊆ R {\displaystyle E\subseteq \mathbb {R} } , the Lebesgue outer measure λ ∗ ( E ) {\displaystyle \lambda ^{\!*\!}(E)} is defined as an infimum...

In mathematics, a metric outer measure is an outer measure μ defined on the subsets of a given metric space (X, d) such that μ ( A ∪ B ) = μ ( A ) + μ...

on all compact sets, outer regular on all Borel sets, and inner regular on open sets. These conditions guarantee that the measure is "compatible" with...

In mathematics, an outer measure μ on n-dimensional Euclidean space Rn is called a Borel regular measure if the following two conditions hold: Every Borel...

specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that assigns a number in [0,∞] to each set...

Lebesgue measure Lorentz space Lifting theory Measurable cardinal Measurable function Minkowski content Outer measure Product measure Pushforward measure Regular...

via the Lebesgue measure. In the one-dimensional case, the Lebesgue outer measure of a set is defined in terms of the lengths of open intervals. Concretely...

Borel measure μg on [a, b] which agrees with w on every interval I. The measure μg arises from an outer measure (in fact, a metric outer measure) given...

probability measure that is neither inner regular nor outer regular. Borel regular measure Radon measure Regularity theorem for Lebesgue measure Ambrosio...

example, Rogers (1998) uses "measure" where this article uses the term "outer measure". Outer measures are not, in general, measures, since they may fail to...

points have measure 0 {\displaystyle 0} , so the measure of any compact subset of this vertical segment is 0 {\displaystyle 0} . But, using outer regularity...

This page is a list of the episodes of The Outer Limits, a 1995 science fiction/dark fantasy television series. The series was broadcast on Showtime from...

outer regular. a Borel regular measure if it is a Borel measure that is also regular. a Radon measure if it is a regular and locally finite measure....

complete measure (or, more precisely, a complete measure space) is a measure space in which every subset of every null set is measurable (having measure zero)...

^{*}:{\mathcal {P}}(\mathbb {R} ^{n})\to [0,\infty ]} denote the Lebesgue outer measure on R n , {\displaystyle \mathbb {R} ^{n},} where P ( R n ) {\displaystyle...

is an outer measure on the family of all subsets. Therefore, it becomes a measure when restricted to the measurable subsets for the outer measure, which...

the inner measure of B {\displaystyle B} equals the outer measure. The common value of the two measures is then simply called the Jordan measure of B {\displaystyle...

first the d-dimensional Hausdorff measure, a fractional-dimension analogue of the Lebesgue measure. First, an outer measure is constructed: Let X {\displaystyle...

Lebesgue measure Lebesgue integration Lebesgue's density theorem Counting measure Complete measure Haar measure Outer measure Borel regular measure Radon...

Equivalently, a cardinal number α is an Ulam number if whenever ν is an outer measure on a set Y, and F a set of pairwise disjoint subsets of Y, ν(⋃F) < ∞...

Xn of a common set X, where X has a Carathéodory outer measure and each Xi has finite outer measure. Their first general formulation is as follows: for...

A tape measure or measuring tape is a flexible ruler used to measure length or distance. It consists of a ribbon of cloth, plastic, fibre glass, or metal...

equivalent. If a Borel measure μ {\displaystyle \mu } is both inner regular and outer regular, it is called a regular Borel measure. If μ {\displaystyle...

measures are often used in combination with outer measures to extend a measure to a larger σ-algebra. If μ {\displaystyle \mu } is a finite measure defined...

diagonal has measure infinity. Outer measure: the proof of Carathéodory's extension theorem is based upon the outer measure concept. Loeb measures, constructed...

semantics of evaluation Valuation (measure theory), a tool for constructing outer measures Valuation (ethics), the determination of the ethic or philosophic value...

sphere of directions. The measure-category duality provides a measure analogue of Luzin sets – sets of positive outer measure, every uncountable subset...

Non-measurable set – Set which cannot be assigned a meaningful "volume" Outer measure – Mathematical function Infinite parity function – Boolean function...

Dimension and outer measure from 1919 is particularly outstanding. In this work, the concepts were introduced which are now known as Hausdorff measure and the...