Displaying similar documents to “Marczewski-Burstin-like characterizations of σ-algebras, ideals, and measurable functions”

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...

Implicit Markov kernels in probability theory

Daniel Hlubinka (2002)

Commentationes Mathematicae Universitatis Carolinae

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Having Polish spaces 𝕏 , 𝕐 and ℤ we shall discuss the existence of an 𝕏 × 𝕐 -valued random vector ( ξ , η ) such that its conditional distributions K x = ℒ ( η ∣ ξ = x ) satisfy e ( x , K x ) = c ( x ) or e ( x , K x ) ∈ C ( x ) for some maps e : 𝕏 × ℳ 1 ( 𝕐 ) → ℤ , c : 𝕏 → ℤ or multifunction C : 𝕏 → 2 ℤ respectively. The problem is equivalent to the existence of universally measurable Markov kernel K : 𝕏 → ℳ 1 ( 𝕐 ) defined implicitly by e ( x , K x ) = c ( x ) or e ( x , K x ) ∈ C ( x ) respectively. In the paper we shall provide sufficient conditions for the existence of the desired Markov kernel. We shall discuss some special solutions of the ( e , c ) - or ( e , C ) -problem...

Exceptional directions for Sierpiński's nonmeasurable sets

B. Kirchheim, Tomasz Natkaniec (1992)

Fundamenta Mathematicae

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In [2] the question was considered in how many directions can a nonmeasurable plane set behave even "better" than the classical one constructed by Sierpiński in [6], in the sense that any line in a given direction intersects the set in at most one point. We considerably improve these results and give a much sharper estimate for the size of the sets of those "better" directions.

A Sard type theorem for Borel mappings

Piotr Hajłasz (1994)

Colloquium Mathematicae

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We find a condition for a Borel mapping f : ℝ m → ℝ n which implies that the Hausdorff dimension of f - 1 ( y ) is less than or equal to m-n for almost all y ∈ ℝ n .

Continuous-, derivative-, and differentiable-restrictions of measurable functions

Jack Brown (1992)

Fundamenta Mathematicae

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We review the known facts and establish some new results concerning continuous-restrictions, derivative-restrictions, and differentiable-restrictions of Lebesgue measurable, universally measurable, and Marczewski measurable functions, as well as functions which have the Baire properties in the wide and restricted senses. We also discuss some known examples and present a number of new examples to show that the theorems are sharp.