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...
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subset E ⊆ R {\displaystyle E\subseteq \mathbb {R} } , the Lebesgue outer measure λ ∗ ( E ) {\displaystyle \lambda ^{\!*\!}(E)} is defined as an infimum...
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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...
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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...
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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...
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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...
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Lebesgue measure Lorentz space Lifting theory Measurable cardinal Measurable function Minkowski content Outer measure Product measure Pushforward measure Regular...
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Length (section Measure theory)
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...
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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...
31 KB (5,360 words) - 23:32, 29 December 2023
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...
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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...
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Lebesgue–Stieltjes integration (redirect from Lebesgue-Stieltjes measure)
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...
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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...
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Set function (category Measures (measure theory))
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....
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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)...
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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...
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finite measure, E ∩ A {\displaystyle E\cap A} is measurable. σ {\displaystyle \sigma } -finite measures and measures arising as the restriction of outer measures...
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Carathéodory's criterion (category Measure theory)
^{*}:{\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...
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Measurable cardinal (category Measures (set theory))
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) < ∞...
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first the d-dimensional Hausdorff measure, a fractional-dimension analogue of the Lebesgue measure. First, an outer measure is constructed: Let X {\displaystyle...
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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...
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Ham sandwich theorem (category Theorems in measure theory)
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...
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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...
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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...
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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...
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Lebesgue measure Lebesgue integration Lebesgue's density theorem Counting measure Complete measure Haar measure Outer measure Borel regular measure Radon...
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Carathéodory's extension theorem (category Theorems in measure theory)
diagonal has measure infinity. Outer measure: the proof of Carathéodory's extension theorem is based upon the outer measure concept. Loeb measures, constructed...
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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...
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Vitali set (category Measure theory)
Non-measurable set – Set which cannot be assigned a meaningful "volume" Outer measure – Mathematical function Infinite parity function – Boolean function...
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Solar System (redirect from Outer planets)
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...
220 KB (21,683 words) - 15:22, 19 August 2024