A Symmetric Density Property for Measurable Sets

Freiling,; Rinne,

doi:10.2307/44153638

Cited by 4 publications

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“…As we have already mentioned, the approximate symmetric basis B ap.s does not have the partitioning property. An extensive proof of D. Preiss and B. Thomson [43] (other proof was given in [18]), leads however to the following result which one may find as a weaker version of partitioning property: for any β s ∆ ∈ B ap.s there exists a set N ⊂ R of measure zero such that for every interval…”

Section: Approximate Symmetric Kurzweil-henstock Integralmentioning

confidence: 99%

On approximately continuous integrals (a survey)

Скворцов¹,

Sworowska²,

Sworowski³

2013

Traditional and Present-Day Topics in Real Analysis. Dedicated to Professor Jan Stanisław Lipiński

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Section: Approximate Symmetric Kurzweil-henstock Integralmentioning

confidence: 99%

On approximately continuous integrals (a survey)

Скворцов¹,

Sworowska²,

Sworowski³

2013

Traditional and Present-Day Topics in Real Analysis. Dedicated to Professor Jan Stanisław Lipiński

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Variations and variational measures in integration theory and some applications

Скворцов¹

1998

J Math Sci

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In this survey, we consider the notions of variation and of variational measure with respect to a derivation basis and explore their feasibility by applying them to the development of an integration theory wide enough to cover many classical problems of analysis.A concept fundamental for the whole theory is that of variational equivalence. A similar notion called "differential equivalence" and a general idea of exploiting it in defining an integral is due to Kolmogorov. It was presented in his classical paper [73]. Later the idea reappeared in the Kurzweil-Henstock theory of the generalized Riemann integral and the related variational integral (see [64,65,74]). This integral has a simpler definition than Lebesgue integral and yet covers a wider field, being equivalent to the Denjoy-Perron integral in the one-dimensional case. These original papers written in the late fifties gave rise to a general theory of nonabsolutely convergent integrals.Here our concern is to present the basic language and methods of the Henstock theory with a view to unifying a variety of approaches to some problems of analysis which require integration processes more powerful than Lebesgue integration. In particular, we examine an application of generalized integrals to the problem of recovering the coefficients of an orthogonal series from its sum by generalized Fourier formulas and some applications to differential equations. We also mention the relation of the Henstock theory to the traditional theory of nonabsolute integrals as it appears, for example, in Saks' book [104].The survey is based mainly on papers referred to in Referativnyi Zhurnal "Matematika" during the last 10-15 years. We also touch on some new results which have not yet appeared. A vast bibliography related to the papers on the theory of nonabsolute integrals published before 1970 can be found in our previous survey [4].

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A new proof of uniqueness for multiple trigonometric series

Ash¹

1989

Proc. Amer. Math. Soc.

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A Symmetric Density Property for Measurable Sets

Cited by 4 publications

References 0 publications

On approximately continuous integrals (a survey)

On approximately continuous integrals (a survey)

Variations and variational measures in integration theory and some applications

A new proof of uniqueness for multiple trigonometric series

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