Displaying similar documents to “Closed convex hull of set of measurable functions, Riemann-measurable functions and measurability of translations”

A problem with almost everywhere equality

Piotr Niemiec (2012)

Annales Polonici Mathematici

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A topological space Y is said to have (AEEP) if the following condition is satisfied: Whenever (X,) is a measurable space and f,g: X → Y are two measurable functions, then the set Δ(f,g) = x ∈ X: f(x) = g(x) is a member of . It is shown that a metrizable space Y has (AEEP) iff the cardinality of Y is not greater than 2 .

Measurable envelopes, Hausdorff measures and Sierpiński sets

Márton Elekes (2003)

Colloquium Mathematicae

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We show that the existence of measurable envelopes of all subsets of ℝⁿ with respect to the d-dimensional Hausdorff measure (0 < d < n) is independent of ZFC. We also investigate the consistency of the existence of d -measurable Sierpiński sets.

Semicontinuous integrands as jointly measurable maps

Oriol Carbonell-Nicolau (2014)

Commentationes Mathematicae Universitatis Carolinae

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Suppose that ( X , 𝒜 ) is a measurable space and Y is a metrizable, Souslin space. Let 𝒜 u denote the universal completion of 𝒜 . For x X , let f ̲ ( x , · ) be the lower semicontinuous hull of f ( x , · ) . If f : X × Y ¯ is ( 𝒜 u ( Y ) , ( ¯ ) ) -measurable, then f ̲ is ( 𝒜 u ( Y ) , ( ¯ ) ) -measurable.

Parametrization of Riemann-measurable selections for multifunctions of two variables with application to differential inclusions

Giovanni Anello, Paolo Cubiotti (2004)

Annales Polonici Mathematici

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We consider a multifunction F : T × X 2 E , where T, X and E are separable metric spaces, with E complete. Assuming that F is jointly measurable in the product and a.e. lower semicontinuous in the second variable, we establish the existence of a selection for F which is measurable with respect to the first variable and a.e. continuous with respect to the second one. Our result is in the spirit of [11], where multifunctions of only one variable are considered.

On a result of K. P. Hart about non-existence of measurable solutions to the discrete expectation maximization problem

Vladimir G. Pestov (2023)

Commentationes Mathematicae Universitatis Carolinae

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It was shown that there is a statistical learning problem – a version of the expectation maximization (EMX) problem – whose consistency in a domain of cardinality continuum under the family of purely atomic probability measures and with finite hypotheses is equivalent to a version of the continuum hypothesis, and thus independent of ZFC. K. P. Hart had subsequently proved that no solution to the EMX problem can be Borel measurable with regard to an uncountable standard Borel structure...

Completely continuous multipliers from L 1 ( G ) into L ( G )

G. Crombez, Willy Govaerts (1984)

Annales de l'institut Fourier

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For a locally compact Hausdorff group G we investigate what functions in L ( G ) give rise to completely continuous multipliers T g from L 1 ( G ) into L ( G ) . In the case of a metrizable group we obtain a complete description of such functions. In particular, for G compact all g in L ( G ) induce completely continuous T g .

The Christensen measurable solutions of a generalization of the Gołąb-Schinzel functional equation

Janusz Brzdęk (1996)

Annales Polonici Mathematici

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Let K denote the set of all reals or complex numbers. Let X be a topological linear separable F-space over K. The following generalization of the result of C. G. Popa [16] is proved. Theorem. Let n be a positive integer. If a Christensen measurable function f: X → K satisfies the functional equation f ( x + f ( x ) n y ) = f ( x ) f ( y ) , then it is continuous or the set x ∈ X : f(x) ≠ 0 is a Christensen zero set.

Universally measurable sets in generic extensions

Paul Larson, Itay Neeman, Saharon Shelah (2010)

Fundamenta Mathematicae

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A subset of a topological space is said to be universally measurable if it is measured by the completion of each countably additive σ-finite Borel measure on the space, and universally null if it has measure zero for each such atomless measure. In 1908, Hausdorff proved that there exist universally null sets of real numbers of cardinality ℵ₁, and thus that there exist at least 2 such sets. Laver showed in the 1970’s that consistently there are just continuum many universally null sets...

Normal integrands and related classes of functions

Anna Kucia, Andrzej Nowak (1995)

Commentationes Mathematicae Universitatis Carolinae

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Let D T × X , where T is a measurable space, and X a topological space. We study inclusions between three classes of extended real-valued functions on D which are upper semicontinuous in x and satisfy some measurability conditions.