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Displaying similar documents to “A Sard type theorem for Borel mappings”

A converse of the Arsenin–Kunugui theorem on Borel sets with σ-compact sections

P. Holický, Miroslav Zelený (2000)

Fundamenta Mathematicae

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Let f be a Borel measurable mapping of a Luzin (i.e. absolute Borel metric) space L onto a metric space M such that f(F) is a Borel subset of M if F is closed in L. We show that then f - 1 ( y ) is a K σ set for all except countably many y ∈ M, that M is also Luzin, and that the Borel classes of the sets f(F), F closed in L, are bounded by a fixed countable ordinal. This gives a converse of the classical theorem of Arsenin and Kunugui. As a particular case we get Taĭmanov’s theorem saying that the...

Very small sets

Haim Judah, Amiran Lior, Ireneusz Recław (1997)

Colloquium Mathematicae

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Necessary condition for measures which are ( L q , L p ) multipliers

Bérenger Akon Kpata, Ibrahim Fofana, Konin Koua (2009)

Annales mathématiques Blaise Pascal

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Let G be a locally compact group and ρ the left Haar measure on G . Given a non-negative Radon measure μ , we establish a necessary condition on the pairs q , p for which μ is a multiplier from L q G , ρ to L p G , ρ . Applied to n , our result is stronger than the necessary condition established by Oberlin in [14] and is closely related to a class of measures defined by Fofana in [7]. When G ...

Generalized projections of Borel and analytic sets

Marek Balcerzak (1996)

Colloquium Mathematicae

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For a σ-ideal I of sets in a Polish space X and for A ⊆ X 2 , we consider the generalized projection (A) of A given by (A) = x ∈ X: Ax ∉ I, where A x =y ∈ X: 〈x,y〉∈ A. We study the behaviour of with respect to Borel and analytic sets in the case when I is a 2 0 -supported σ-ideal. In particular, we give an alternative proof of the recent result of Kechris showing that [ 1 1 ( X 2 ) ] = 1 1 ( X ) for a wide class of 2 0 -supported σ-ideals.

Vitali sets and Hamel bases that are Marczewski measurable

Arnold Miller, Strashimir Popvassilev (2000)

Fundamenta Mathematicae

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We give examples of a Vitali set and a Hamel basis which are Marczewski measurable and perfectly dense. The Vitali set example answers a question posed by Jack Brown. We also show there is a Marczewski null Hamel basis for the reals, although a Vitali set cannot be Marczewski null. The proof of the existence of a Marczewski null Hamel basis for the plane is easier than for the reals and we give it first. We show that there is no easy way to get a Marczewski null Hamel basis for the reals...

Multifractals and projections.

Fadhila Bahroun, Imen Bhouri (2006)

Extracta Mathematicae

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In this paper, we generalize the result of Hunt and Kaloshin [5] about the L-spectral dimensions of a measure and that of its projections. The results we obtain, allow to study an untreated case in their work and to find a relationship between the multifractal spectrum of a measure and that of its projections.

Lebesgue measure and mappings of the Sobolev class W 1 , n

O. Martio (1995)

Banach Center Publications

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We present a survey of the Lusin condition (N) for W 1 , n -Sobolev mappings f : G n defined in a domain G of n . Applications to the boundary behavior of conformal mappings are discussed.

Functions characterized by images of sets

Krzysztof Ciesielski, Dikran Dikrajan, Stephen Watson (1998)

Colloquium Mathematicae

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For non-empty topological spaces X and Y and arbitrary families 𝒜 𝒫 ( X ) and 𝒫 ( Y ) we put 𝒞 𝒜 , =f ∈ Y X : (∀ A ∈ 𝒜 )(f[A] ∈ ) . We examine which classes of functions Y X can be represented as 𝒞 𝒜 , . We are mainly interested in the case when = 𝒞 ( X , Y ) is the class of all continuous functions from X into Y. We prove that for a non-discrete Tikhonov space X the class = 𝒞 (X,ℝ) is not equal to 𝒞 𝒜 , for any 𝒜 𝒫 ( X ) and 𝒫 (ℝ). Thus, 𝒞 (X,ℝ) cannot be characterized by images of sets. We also show that none of the following...