On the extremality of regular extensions of contents and measures

Wolfgang Adamski

Displaying similar documents to “On the extremality of regular extensions of contents and measures”

Regularity of certain sets in ℂⁿ

Nguyen Quang Dieu (2003)

Annales Polonici Mathematici

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A subset K of ℂⁿ is said to be regular in the sense of pluripotential theory if the pluricomplex Green function (or Siciak extremal function) $V_{K}$ is continuous in ℂⁿ. We show that K is regular if the intersections of K with sufficiently many complex lines are regular (as subsets of ℂ). A complete characterization of regularity for Reinhardt sets is also given.

Order convergence of vector measures on topological spaces

Surjit Singh Khurana (2008)

Mathematica Bohemica

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Let $X$ be a completely regular Hausdorff space, $E$ a boundedly complete vector lattice, $C_{b} (X)$ the space of all, bounded, real-valued continuous functions on $X$ , $ℱ$ the algebra generated by the zero-sets of $X$ , and $μ C_{b} (X) \to E$ a positive linear map. First we give a new proof that $μ$ extends to a unique, finitely additive measure $μ ℱ \to E^{+}$ such that $ν$ is inner regular by zero-sets and outer regular by cozero sets. Then some order-convergence theorems about nets of $E^{+}$ -valued finitely additive measures on $ℱ$ are proved,...

Extremal plurisubharmonic functions in $C^{N}$

Józef Siciak (1981)

Annales Polonici Mathematici

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Normal integral bases and tameness conditions for Kummer extensions

Ilaria Del Corso, Lorenzo Paolo Rossi (2013)

Acta Arithmetica

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We present a detailed analysis of some properties of a general tamely ramified Kummer extension of number fields L/K. Our main achievement is a criterion for the existence of a normal integral basis for a general Kummer extension, which generalizes the existing results. Our approach also allows us to explicitly describe the Steinitz class of L/K and we get an easy criterion for this class to be trivial. In the second part of the paper we restrict to the particular case of tame Kummer...

Mean values and associated measures of $δ$ -subharmonic functions

Neil A. Watson (2002)

Mathematica Bohemica

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Let $u$ be a $δ$ -subharmonic function with associated measure $μ$ , and let $v$ be a superharmonic function with associated measure $ν$ , on an open set $E$ . For any closed ball $B (x, r)$ , of centre $x$ and radius $r$ , contained in $E$ , let $ℳ (u, x, r)$ denote the mean value of $u$ over the surface of the ball. We prove that the upper and lower limits as $s, t \to 0$ with $0 < s < t$ of the quotient $(ℳ (u, x, s) - ℳ (u, x, t)) / (ℳ (v, x, s) - ℳ (v, x, t))$ , lie between the upper and lower limits as $r \to 0 +$ of the quotient $μ (B (x, r)) / ν (B (x, r))$ . This enables us to use some well-known measure-theoretic results to prove new variants...

Existence and integral representation of regular extensions of measures

Werner Rinkewitz (2001)

Colloquium Mathematicae

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Let ℒ be a δ-lattice in a set X, and let ν be a measure on a sub-σ-algebra of σ(ℒ). It is shown that ν extends to an ℒ-regular measure on σ(ℒ) provided ν*|ℒ is σ-smooth at ∅ and ν*(L) = inf ν*(U)|X ∖ U ∈ ℒ, Usupset L for all L ∈ ℒ. Moreover, a Choquet type representation theorem is proved for the set of all such extensions.

Relative Bogomolov extensions

Robert Grizzard (2015)

Acta Arithmetica

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A subfield K ⊆ ℚ̅ has the Bogomolov property if there exists a positive ε such that no non-torsion point of $K^{\times}$ has absolute logarithmic height below ε. We define a relative extension L/K to be Bogomolov if this holds for points of $L^{\times} ∖ K^{\times}$ . We construct various examples of extensions which are and are not Bogomolov. We prove a ramification criterion for this property, and use it to show that such extensions can always be constructed if some rational prime has bounded ramification index in K. ...

A general upper bound in extremal theory of sequences

Martin Klazar (1992)

Commentationes Mathematicae Universitatis Carolinae

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We investigate the extremal function $f (u, n)$ which, for a given finite sequence $u$ over $k$ symbols, is defined as the maximum length $m$ of a sequence $v = a_{1} a_{2} . . . a_{m}$ of integers such that 1) $1 \leq a_{i} \leq n$ , 2) $a_{i} = a_{j}, i \neq j$ implies $| i - j | \geq k$ and 3) $v$ contains no subsequence of the type $u$ . We prove that $f (u, n)$ is very near to be linear in $n$ for any fixed $u$ of length greater than 4, namely that $f (u, n) = O (n 2^{O (α {(n)}^{| u | - 4})}) .$ Here $| u |$ is the length of $u$ and $α (n)$ is the inverse to the Ackermann function and goes to infinity very slowly. This result extends the estimates in [S] and...

Compactness and extreme points of the set of quasi-measure extensions of a quasi-measure

Zbigniew Lipecki

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The memoir is based on a series of six papers by the author published over the years 1995-2007. It continues the work of D. Plachky (1970, 1976). It also owes some inspiration, among others, to papers by J. Łoś and E. Marczewski (1949), D. Bierlein and W. J. A. Stich (1989), D. Bogner and R. Denk (1994), and A. Ülger (1996). Let and ℜ be algebras of subsets of a set Ω with ⊂ ℜ. Given a quasi-measure μ on , i.e., μ ∈ ba₊(), we denote by E(μ) the convex set of all quasi-measure extensions...