Displaying similar documents to “Universally measurable sets in generic extensions”

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...

Closed convex hull of set of measurable functions, Riemann-measurable functions and measurability of translations

Michel Talagrand (1982)

Annales de l'institut Fourier

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Let G be a locally compact group. Let L t be the left translation in L ∞ ( G ) , given by L t f ( x ) = f ( t x ) . We characterize (undre a mild set-theoretical hypothesis) the functions f ∈ L ∞ ( G ) such that the map t → L t f from G into L ∞ ( G ) is scalarly measurable (i.e. for φ ∈ L ∞ ( G ) * , t → φ ( L t f ) is measurable). We show that it is the case when t → θ ( L f t ) is measurable for each character θ , and if G is compact, if and only if f is Riemann-measurable. We show that t → L t f is Borel measurable if and only if f is left uniformly continuous. Some of the measure-theoretic...

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 ℋ d -measurable Sierpiński sets.

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 ℵ ₀ .

Bernoulli sequences and Borel measurability in ( 0 , 1 )

Petr Veselý (1993)

Commentationes Mathematicae Universitatis Carolinae

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The necessary and sufficient condition for a function f : ( 0 , 1 ) → [ 0 , 1 ] to be Borel measurable (given by Theorem stated below) provides a technique to prove (in Corollary 2) the existence of a Borel measurable map H : { 0 , 1 } ℕ → { 0 , 1 } ℕ such that ℒ ( H ( X p ) ) = ℒ ( X 1 / 2 ) holds for each p ∈ ( 0 , 1 ) , where X p = ( X 1 p , X 2 p , ... ) denotes Bernoulli sequence of random variables with P [ X i p = 1 ] = p .

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 → ℝ ¯ is ( 𝒜 u ⊗ ℬ ( Y ) , ℬ ( ℝ ¯ ) ) -measurable, then f ̲ is ( 𝒜 u ⊗ ℬ ( Y ) , ℬ ( ℝ ¯ ) ) -measurable.

When is ℝ the union of an increasing family of null sets?

Juan González-Hernández, Fernando Hernández-Hernández, César E. Villarreal (2007)

Commentationes Mathematicae Universitatis Carolinae

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We study the problem in the title and show that it is equivalent to the fact that every set of reals is an increasing union of measurable sets. We also show the relationship of it with Sierpi'nski sets.