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...

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...

inner regular. A measure is called outer regular if every measurable set is outer regular. A measure is called regular if it is outer regular and inner...

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...

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...

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)...

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...

finite measure, E ∩ A {\displaystyle E\cap A} is measurable. σ {\displaystyle \sigma } -finite measures and measures arising as the restriction of outer measures...

^{*}:{\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...

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) < ∞...

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

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...

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...

Outer space (or simply space) is the expanse that exists beyond Earth's atmosphere and between celestial bodies. It contains ultra-low levels of particle...

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

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

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...

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

fusion of hydrogen into helium at its core, releasing this energy from its outer photosphere. Astronomers classify it as a G-type main-sequence star. The...