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

K. P. S. Bhaskara Rao; Vincenzo Aversa

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

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

Euline Green (1971)

Colloquium Mathematicae

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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.

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.

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. ...

On the Behavior of Power Series with Completely Additive Coefficients

Oleg Petrushov (2015)

Bulletin of the Polish Academy of Sciences. Mathematics

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Consider the power series $(z) = \sum_{n = 1}^{\infty} α (n) z ⁿ$ , where α(n) is a completely additive function satisfying the condition α(p) = o(lnp) for prime numbers p. Denote by e(l/q) the root of unity $e^{2 π i l / q}$ . We give effective omega-estimates for $(e (l / p^{k}) r)$ when r → 1-. From them we deduce that if such a series has non-singular points on the unit circle, then it is a zero function.

The number of squares and $B_{h} [g]$ sets

Ben Green (2001)

Acta Arithmetica

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On the sum of dilations of a set

Antal Balog, George Shakan (2014)

Acta Arithmetica

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We show that for any relatively prime integers 1 ≤ p < q and for any finite A ⊂ ℤ one has ${| p \cdot A + q \cdot A | \geq (p + q) | A | - (p q)}^{(p + q - 3) (p + q) + 1}$ .

$F_{σ}$ -additive covers of Čech complete and scattered-K-analytic spaces

Jiří Spurný (2008)

Fundamenta Mathematicae

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We prove that an $F_{σ}$ -additive cover of a Čech complete, or more generally scattered-K-analytic space, has a σ-scattered refinement. This generalizes results of G. Koumoullis and R. W. Hansell.

$L_{1}$ contains every two-dimensional normed space

David Yost (1988)

Annales Polonici Mathematici

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Spectra of elements in the group ring of SU(2)

Alex Gamburd, Dmitry Jakobson, Peter Sarnak (1999)

Journal of the European Mathematical Society

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We present a new method for establishing the ‘‘gap” property for finitely generated subgroups of $SU (2)$ , providing an elementary solution of Ruziewicz problem on $S^{2}$ as well as giving many new examples of finitely generated subgroups of $SU (2)$ with an explicit gap. The distribution of the eigenvalues of the elements of the group ring $𝐑 [SU (2)]$ in the $N$ -th irreducible representation of $SU (2)$ is also studied. Numerical experiments indicate that for a generic (in measure) element of $𝐑 [SU (2)]$ , the “unfolded” consecutive...

A generalization of a theorem of Erdős-Rényi to m-fold sums and differences

Kathryn E. Hare, Shuntaro Yamagishi (2014)

Acta Arithmetica

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Let m ≥ 2 be a positive integer. Given a set E(ω) ⊆ ℕ we define $r_{N}^{(m)} (ω)$ to be the number of ways to represent N ∈ ℤ as a combination of sums and differences of m distinct elements of E(ω). In this paper, we prove the existence of a “thick” set E(ω) and a positive constant K such that $r_{N}^{(m)} (ω) < K$ for all N ∈ ℤ. This is a generalization of a known theorem by Erdős and Rényi. We also apply our results to harmonic analysis, where we prove the existence of certain thin sets.

A theorem on isotropic spaces

Félix Cabello Sánchez (1999)

Studia Mathematica

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Let X be a normed space and $G_{F} (X)$ the group of all linear surjective isometries of X that are finite-dimensional perturbations of the identity. We prove that if $G_{F} (X)$ acts transitively on the unit sphere then X must be an inner product space.

On the concentration of certain additive functions

Dimitris Koukoulopoulos (2014)

Acta Arithmetica

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We study the concentration of the distribution of an additive function f when the sequence of prime values of f decays fast and has good spacing properties. In particular, we prove a conjecture by Erdős and Kátai on the concentration of $f (n) = \sum_{p | n} {(l o g p)}^{- c}$ when c > 1.

Effective results for Diophantine equations over finitely generated domains

Attila Bérczes, Jan-Hendrik Evertse, Kálmán Győry (2014)

Acta Arithmetica

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Let A be an arbitrary integral domain of characteristic 0 that is finitely generated over ℤ. We consider Thue equations F(x,y) = δ in x,y ∈ A, where F is a binary form with coefficients from A, and δ is a non-zero element from A, and hyper- and superelliptic equations $f (x) = δ y^{m}$ in x,y ∈ A, where f ∈ A[X], δ ∈ A∖0 and $m \in ℤ_{\geq 2}$ . Under the necessary finiteness conditions we give effective upper bounds for the sizes of the solutions of the equations in terms of appropriate representations for A, δ, F,...

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...

Some properties and applications of equicompact sets of operators

E. Serrano, C. Piñeiro, J. M. Delgado (2007)

Studia Mathematica

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Let X and Y be Banach spaces. A subset M of (X,Y) (the vector space of all compact operators from X into Y endowed with the operator norm) is said to be equicompact if every bounded sequence (xₙ) in X has a subsequence $(x_{k (n)}) ₙ$ such that $(T x_{k (n)}) ₙ$ is uniformly convergent for T ∈ M. We study the relationship between this concept and the notion of uniformly completely continuous set and give some applications. Among other results, we obtain a generalization of the classical Ascoli theorem and a compactness...

On the number of finite algebraic structures

Erhard Aichinger, Peter Mayr, R. McKenzie (2014)

Journal of the European Mathematical Society

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We prove that every clone of operations on a finite set $A$ , if it contains a Malcev operation, is finitely related – i.e., identical with the clone of all operations respecting $R$ for some finitary relation $R$ over $A$ . It follows that for a fixed finite set $A$ , the set of all such Malcev clones is countable. This completes the solution of a problem that was first formulated in 1980, or earlier: how many Malcev clones can finite sets support? More generally, we prove that every finite algebra...

$ω$ -compact elements of normed algebras

Gh. Constantin (1973)

Matematički Vesnik

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Displaying similar documents to “A remark on E. Green’s paper “Completeness of L p -spaces over finitely additive set functions””

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