Why $\lambda $-additive (fuzzy) measures?

Ion Chiţescu

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

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 $a_{n} \leq b_{m}$ for all $n, m$ , there exists an element $b$ in $𝒜$ such that $a_{n} \leq b \leq b_{n}$ for all $n$ . Let $𝒜$ denote an algebra with the property (I). It is shown that if $(μ_{n} : 𝒜 \to X)$ ( $X$ a Banach space) is a sequence of strongly additive measures such that ${lim}_{n} μ_{n} (a)$ exists for each $a \in 𝒜$ , 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 $X \subseteq 2^{ω}$ , 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.

Addendum to “On Meager Additive and Null Additive Sets in the Cantor space $2^{ω}$ and in ℝ” (Bull. Polish Acad. Sci. Math. 57 (2009), 91-99)

Tomasz Weiss (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

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We prove in ZFC that there is a set $A \subseteq 2^{ω}$ and a surjective function H: A → ⟨0,1⟩ such that for every null additive set X ⊆ ⟨0,1), $H^{- 1} (X)$ is null additive in $2^{ω}$ . This settles in the affirmative a question of T. Bartoszyński.

On vector valued measure spaces of bounded $Φ$ -variation containing copies of $ℓ_{\infty}$

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 $ℓ_{\infty}$ 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.

Representation of operators on $L^{1}$ -spaces by nondirect product measures

Anzelm Iwanik (1982)

Colloquium Mathematicum

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On Ordinary and Standard Lebesgue Measures on $ℝ^{\infty}$

Gogi Pantsulaia (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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New concepts of Lebesgue measure on $ℝ^{\infty}$ 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 $ℝ^{\infty}$ for α = (1,1,...).

Completeness of $L^{p}$ -spaces over finitely additive set functions

Euline Green (1971)

Colloquium Mathematicae

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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 $\sum_{x \leq n \leq x + H} 1 / f (Q (n))$ and $\sum_{x \leq p \leq x + H} 1 / f (Q (p))$ , 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.

$S$ -measures, $T$ -measures and distinguished classes of fuzzy measures

Peter Struk, Andrea Stupňanová (2006)

Kybernetika

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$S$ -measures are special fuzzy measures decomposable with respect to some fixed t-conorm $S$ . We investigate the relationship of $S$ -measures with some distinguished properties of fuzzy measures, such as subadditivity, submodularity, belief, etc. We show, for example, that each $S_{P}$ -measure is a plausibility measure, and that each $S$ -measure is submodular whenever $S$ is 1-Lipschitz.

A remark on E. Green’s paper “Completeness of $L^{p}$ -spaces over finitely additive set functions”

K. P. S. Bhaskara Rao, Vincenzo Aversa (1987)

Colloquium Mathematicae

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A note on the super-additive and sub-additive transformations of aggregation functions: The multi-dimensional case

Fateme Kouchakinejad, Alexandra Šipošová (2017)

Kybernetika

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For an aggregation function $A$ we know that it is bounded by $A^{*}$ and $A_{*}$ which are its super-additive and sub-additive transformations, respectively. Also, it is known that if $A^{*}$ is directionally convex, then $A = A^{*}$ and $A_{*}$ is linear; similarly, if $A_{*}$ is directionally concave, then $A = A_{*}$ and $A^{*}$ is linear. We generalize these results replacing the directional convexity and concavity conditions by the weaker assumptions of overrunning a super-additive function and underrunning a sub-additive function, respectively. ...

Continuous linear functionals on the space of Borel vector measures

Pola Siwek (2008)

Annales Polonici Mathematici

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We study properties of the space ℳ of Borel vector measures on a compact metric space X, taking values in a Banach space E. The space ℳ is equipped with the Fortet-Mourier norm ${| | \cdot | |}_{ℱ}$ and the semivariation norm ||·||(X). The integral introduced by K. Baron and A. Lasota plays the most important role in the paper. Investigating its properties one can prove that in most cases the space $(ℳ, | | \cdot | |_{ℱ}) *$ is contained in but not equal to the space (ℳ,||·||(X))*. We obtain a representation of the continuous functionals...

Limit theorems for random fields

Nguyen van Thu

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CONTENTSIntroduction............................................................................................................................................................................ 51. Notation and preliminaries............................................................................................................................................ 52. Statement of the problem..................................................................................................................................................

Simple fractions and linear decomposition of some convolutions of measures

Jolanta K. Misiewicz, Roger Cooke (2001)

Discussiones Mathematicae Probability and Statistics

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Every characteristic function φ can be written in the following way: φ(ξ) = 1/(h(ξ) + 1), where h(ξ) = ⎧ 1/φ(ξ) - 1 if φ(ξ) ≠ 0 ⎨ ⎩ ∞ if φ(ξ) = 0 This simple remark implies that every characteristic function can be treated as a simple fraction of the function h(ξ). In the paper, we consider a class C(φ) of all characteristic functions of the form $φ_{a} (ξ) = [a / (h (ξ) + a)]$ , where φ(ξ) is a fixed characteristic function. Using the well known theorem on simple fraction decomposition of rational functions we obtain...

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

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