## 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

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 ${f}^{-1}\left(y\right)$ is a ${K}_{\sigma }$ 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

Colloquium Mathematicae

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### Very small sets

Colloquium Mathematicae

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### Necessary condition for measures which are $\left({L}^{q},{L}^{p}\right)$ multipliers

Annales mathématiques Blaise Pascal

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Let $G$ be a locally compact group and $\rho$ the left Haar measure on $G$. Given a non-negative Radon measure $\mu$, we establish a necessary condition on the pairs $\left(q,\phantom{\rule{4pt}{0ex}}p\right)$ for which $\mu$ is a multiplier from ${L}^{q}\left(G,\phantom{\rule{4pt}{0ex}}\rho \right)$ to ${L}^{p}\left(G,\phantom{\rule{4pt}{0ex}}\rho \right)$. Applied to ${ℝ}^{n}$, our result is stronger than the necessary condition established by Oberlin in  and is closely related to a class of measures defined by Fofana in . When $G$...

### Generalized projections of Borel and analytic sets

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 [${\sum }_{1}^{1}\left({X}^{2}\right)\right]={\sum }_{1}^{1}\left(X\right)$ for a wide class of ${\sum }_{2}^{0}$-supported σ-ideals.

### Vitali sets and Hamel bases that are Marczewski measurable

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.

Extracta Mathematicae

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In this paper, we generalize the result of Hunt and Kaloshin  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 ${W}^{1,n}$

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

Colloquium Mathematicae

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For non-empty topological spaces X and Y and arbitrary families $𝒜$$𝒫\left(X\right)$ and $ℬ\subseteq 𝒫\left(Y\right)$ we put ${𝒞}_{𝒜,ℬ}$=f ∈ ${Y}^{X}$ : (∀ A ∈ $𝒜$)(f[A] ∈ $ℬ\right)$. We examine which classes of functions $ℱ$${Y}^{X}$ can be represented as ${𝒞}_{𝒜,ℬ}$. We are mainly interested in the case when $ℱ=𝒞\left(X,Y\right)$ 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$$𝒫\left(X\right)$ and $ℬ$$𝒫$(ℝ). Thus, $𝒞$(X,ℝ) cannot be characterized by images of sets. We also show that none of the following...