A Sard type theorem for Borel mappings

Piotr Hajłasz

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...

Some additive properties of special sets of reals

Ireneusz Recław (1991)

Colloquium Mathematicae

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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 $ℝ^{n}$ , 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 $\sum_{2}^{0}$ -supported σ-ideal. In particular, we give an alternative proof of the recent result of Kechris showing that [ $\sum_{1}^{1} (X^{2})] = \sum_{1}^{1} (X)$ for a wide class of $\sum_{2}^{0}$ -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 $f : G \to ℝ^{n}$ defined in a domain G of $ℝ^{n}$ . 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 $ℬ \subseteq 𝒫 (Y)$ we put $𝒞_{𝒜, ℬ}$ =f ∈ $Y^{X}$ : (∀ A ∈ $𝒜$ )(f[A] ∈ $ℬ)$ . We examine which classes of functions $ℱ$ ⊆ $Y^{X}$ can be represented as $𝒞_{𝒜, ℬ}$ . We are mainly interested in the case when $ℱ = 𝒞 (X, Y)$ is the class of all continuous functions from X into Y. We prove that for a non-discrete Tikhonov space X the class $ℱ = 𝒞$ (X,ℝ) is not equal to $𝒞_{𝒜, ℬ}$ for any $𝒜$ ⊆ $𝒫 (X)$ and $ℬ$ ⊆ $𝒫$ (ℝ). Thus, $𝒞$ (X,ℝ) cannot be characterized by images of sets. We also show that none of the following...