On Smital properties

Bartoszewicz, Artur; Filipczak, Małgorzata; Natkaniec, Tomasz

doi:10.1016/j.topol.2011.06.044

Cited by 19 publications

(15 citation statements)

References 7 publications

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“…Hence, by (2), DS (1) I (F ) ⊃ DS (3) I (F ). But, again from DS (2) I (F ) = DS (2) I (F ) and (1), it follows that DS (1) I (F ) ⊂ DS (3) I (F ).…”

Section: The Operator Dsmentioning

confidence: 96%

“…We say that a triple (A, I, τ) has the Smital property if, for any set A ∈ A\I and any dense set P ⊂ X, the set (A + P) (the complement of A + P) belongs to I (see [2]).…”

Section: The Operator Dsmentioning

confidence: 99%

“…Statements (1)-(4) are straightforward consequences of the definition of DS I . For (5), observe that from DS I (F ) = DS I (F ) and (1) we obtain F ⊂ DS (2) I (F ). Hence, by (2), DS (1) I (F ) ⊃ DS (3) I (F ).…”

Section: The Operator Dsmentioning

confidence: 99%

“…Looking for a more general approach, we observe that the σ-field L and the σ-ideal of sets of measure zero can be replaced by any pair (A, I), where A is a field of subsets of X and I ⊂ A is an ideal. In this context, the Smital property was studied in [2]. It is easily seen that no topology is needed to formulate and study the Smital property.…”

Section: Introductionmentioning

confidence: 99%

“…c 2017 Australian Mathematical Publishing Association Inc. 0004-9727/2017 $16.00 [2] Density, Smital property and quasicontinuity 247…”

Section: Introductionmentioning

confidence: 99%

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Density, Smital Property and Quasicontinuity

Horbaczewska

Lindner

2017

Bull. Aust. Math. Soc.

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“…Hence, by (2), DS (1) I (F ) ⊃ DS (3) I (F ). But, again from DS (2) I (F ) = DS (2) I (F ) and (1), it follows that DS (1) I (F ) ⊂ DS (3) I (F ).…”

Section: The Operator Dsmentioning

confidence: 96%

“…We say that a triple (A, I, τ) has the Smital property if, for any set A ∈ A\I and any dense set P ⊂ X, the set (A + P) (the complement of A + P) belongs to I (see [2]).…”

Section: The Operator Dsmentioning

confidence: 99%

Section: The Operator Dsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…c 2017 Australian Mathematical Publishing Association Inc. 0004-9727/2017 $16.00 [2] Density, Smital property and quasicontinuity 247…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Density, Smital Property and Quasicontinuity

Horbaczewska

Lindner

2017

Bull. Aust. Math. Soc.

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Category-measure duality: convexity, midpoint convexity and Berz sublinearity

Bingham

Ostaszewski

2017

Aequat. Math.

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Abstract. Category-measure duality concerns applications of Baire-category methods that have measure-theoretic analogues. The set-theoretic axiom needed in connection with the Baire category theorem is the Axiom of Dependent Choice, DC rather than the Axiom of Choice, AC. Berz used the Hahn-Banach theorem over Q to prove that the graph of a measurable sublinear function that is Q + -homogeneous consists of two half-lines through the origin. We give a category form of the Berz theorem. Our proof is simpler than that of the classical measure-theoretic Berz theorem, our result contains Berz's theorem rather than simply being an analogue of it, and we use only DC rather than AC. Furthermore, the category form easily generalizes: the graph of a Baire sublinear function defined on a Banach space is a cone. The results are seen to be of automatic-continuity type. We use Christensen Haar null sets to extend the category approach beyond the locally compact setting where Haar measure exists. We extend Berz's result from Euclidean to Banach spaces, and beyond. Passing from sublinearity to convexity, we extend the Bernstein-Doetsch theorem and related continuity results, allowing our conditions to be 'local'-holding off some exceptional set.Mathematics Subject Classification. Primary 26A03, 39B62.

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Variants on the Berz sublinearity theorem

Bingham

Ostaszewski

2018

Aequat. Math.

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To Roy O. Davies on his 90 th birthday.Abstract. We consider variants on the classical Berz sublinearity theorem, using only DC, the Axiom of Dependent Choices, rather than AC, the Axiom of Choice which Berz used. We consider thinned versions, in which conditions are imposed on only part of the domain of the function -results of quantifierweakening type. There are connections with classical results on subadditivity. We close with a discussion of the extensive related literature.

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On Smital properties

Cited by 19 publications

References 7 publications

Density, Smital Property and Quasicontinuity

Density, Smital Property and Quasicontinuity

Category-measure duality: convexity, midpoint convexity and Berz sublinearity

Variants on the Berz sublinearity theorem

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