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Displaying similar documents to “Why λ -additive (fuzzy) measures?”

Finitely-additive, countably-additive and internal probability measures

Haosui Duanmu, William Weiss (2018)

Commentationes Mathematicae Universitatis Carolinae

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We discuss two ways to construct standard probability measures, called push-down measures, from internal probability measures. We show that the Wasserstein distance between an internal probability measure and its push-down measure is infinitesimal. As an application to standard probability theory, we show that every finitely-additive Borel probability measure P on a separable metric space is a limit of a sequence of countably-additive Borel probability measures { P n } n in the sense that f d P = lim n f d P n ...

Strong measure zero and meager-additive sets through the prism of fractal measures

Ondřej Zindulka (2019)

Commentationes Mathematicae Universitatis Carolinae

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We develop a theory of sharp measure zero sets that parallels Borel’s strong measure zero, and prove a theorem analogous to Galvin–Mycielski–Solovay theorem, namely that a set of reals has sharp measure zero if and only if it is meager-additive. Some consequences: A subset of 2 ω is meager-additive if and only if it is -additive; if f : 2 ω 2 ω is continuous and X is meager-additive, then so is f ( X ) .

On Vitali-Hahn-Saks-Nikodym type theorems

Barbara T. Faires (1976)

Annales de l'institut Fourier

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A Boolean algebra 𝒜 has the interpolation property (property (I)) if given sequences ( a n ) , ( b m ) in 𝒜 with a n b m for all n , m , there exists an element b in 𝒜 such that a n b b n for all n . Let 𝒜 denote an algebra with the property (I). It is shown that if ( μ n : 𝒜 X ) ( X a Banach space) is a sequence of strongly additive measures such that lim n μ n ( a ) exists for each a 𝒜 , then μ ( a ) = lim n μ n ( a ) defines a strongly additive map from 𝒜 to X the μ n ' s are uniformly strongly additive. The Vitali-Hahn-Saks (VHS) theorem for strongly additive...

On Meager Additive and Null Additive Sets in the Cantor Space 2 ω and in ℝ

Tomasz Weiss (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let T be the standard Cantor-Lebesgue function that maps the Cantor space 2 ω onto the unit interval ⟨0,1⟩. We prove within ZFC that for every X 2 ω , X is meager additive in 2 ω iff T(X) is meager additive in ⟨0,1⟩. As a consequence, we deduce that the cartesian product of meager additive sets in ℝ remains meager additive in ℝ × ℝ. In this note, we also study the relationship between null additive sets in 2 ω and ℝ.

On vector valued measure spaces of bounded Φ -variation containing copies of

María J. Rivera (2001)

Czechoslovak Mathematical Journal

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Given a Young function Φ , we study the existence of copies of c 0 and in c a b v Φ ( μ , X ) and in c a b s v Φ ( μ , X ) , the countably additive, μ -continuous, and X -valued measure spaces of bounded Φ -variation and bounded Φ -semivariation, respectively.

On Ordinary and Standard Lebesgue Measures on

Gogi Pantsulaia (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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New concepts of Lebesgue measure on are proposed and some of their realizations in the ZFC theory are given. Also, it is shown that Baker’s both measures [1], [2], Mankiewicz and Preiss-Tišer generators [6] and the measure of [4] are not α-standard Lebesgue measures on for α = (1,1,...).

Sums of reciprocals of additive functions running over short intervals

J.-M. De Koninck, I. Kátai (2007)

Colloquium Mathematicae

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Letting f(n) = A log n + t(n), where t(n) is a small additive function and A a positive constant, we obtain estimates for the quantities x n x + H 1 / f ( Q ( n ) ) and x p x + H 1 / f ( Q ( p ) ) , where H = H(x) satisfies certain growth conditions, p runs over prime numbers and Q is a polynomial with integer coefficients, whose leading coefficient is positive, and with all its roots simple.