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

Vladimir G. Pestov

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

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 $2^{ℵ ₁}$ such sets. Laver showed in the 1970’s that consistently there are just continuum many universally null 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 $x \in X$ , let $\underset{̲}{f} (x, \cdot)$ be the lower semicontinuous hull of $f (x, \cdot)$ . If $f : X \times Y \to \bar{ℝ}$ is $(𝒜^{u} \otimes ℬ (Y), ℬ (\bar{ℝ}))$ -measurable, then $\underset{̲}{f}$ is $(𝒜^{u} \otimes ℬ (Y), ℬ (\bar{ℝ}))$ -measurable.

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 $2^{ℵ ₀}$ .

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.

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 $L^{\infty} (G)$ , given by $L_{t} f (x) = f (t x)$ . We characterize (undre a mild set-theoretical hypothesis) the functions $f \in L^{\infty} (G)$ such that the map $t \to L_{t} f$ from $G$ into $L^{\infty} (G)$ is scalarly measurable (i.e. for $φ \in L^{\infty} (G)^{*}$ , $t \to φ (L_{t} f)$ is measurable). We show that it is the case when $t \to θ (L_{f} t)$ is measurable for each character $θ$ , and if $G$ is compact, if and only if $f$ is Riemann-measurable. We show that $t \to L_{t} f$ is Borel measurable if and only if $f$ is left uniformly continuous. Some of the measure-theoretic...

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.

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

Can we assign the Borel hulls in a monotone way?

Márton Elekes, András Máthé (2009)

Fundamenta Mathematicae

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A hull of A ⊆ [0,1] is a set H containing A such that λ*(H) = λ*(A). We investigate all four versions of the following problem. Does there exist a monotone (with respect to inclusion) map that assigns a Borel/ $G_{δ}$ hull to every negligible/measurable subset of [0,1]? Three versions turn out to be independent of ZFC, while in the fourth case we only prove that the nonexistence of a monotone $G_{δ}$ hull operation for all measurable sets is consistent. It remains open whether existence here is also...

Homeomorphisms of composants of Knaster continua

Sonja Štimac (2002)

Fundamenta Mathematicae

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The Knaster continuum $K_{p}$ is defined as the inverse limit of the pth degree tent map. On every composant of the Knaster continuum we introduce an order and we consider some special points of the composant. These are used to describe the structure of the composants. We then prove that, for any integer p ≥ 2, all composants of $K_{p}$ having no endpoints are homeomorphic. This generalizes Bandt’s result which concerns the case p = 2.

Decompositions of the plane and the size of the continuum

Ramiro de la Vega (2009)

Fundamenta Mathematicae

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We consider a triple ⟨E₀,E₁,E₂⟩ of equivalence relations on ℝ² and investigate the possibility of decomposing the plane into three sets ℝ² = S₀ ∪ S₁ ∪ S₂ in such a way that each $S_{i}$ intersects each $E_{i}$ -class in finitely many points. Many results in the literature, starting with a famous theorem of Sierpiński, show that for certain triples the existence of such a decomposition is equivalent to the continuum hypothesis. We give a characterization in ZFC of the triples for which the decomposition...

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 $F : T \times X \to 2^{E}$ , 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.

How many normal measures can $ℵ_{ω + 1}$ carry?

Arthur W. Apter (2006)

Fundamenta Mathematicae

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We show that assuming the consistency of a supercompact cardinal with a measurable cardinal above it, it is possible for $ℵ_{ω + 1}$ to be measurable and to carry exactly τ normal measures, where $τ \geq ℵ_{ω + 2}$ is any regular cardinal. This contrasts with the fact that assuming AD + DC, $ℵ_{ω + 1}$ is measurable and carries exactly three normal measures. Our proof uses the methods of [6], along with a folklore technique and a new method due to James Cummings.

Categoricity of theories in $L_{κ , ω}$ , when κ is a measurable cardinal. Part 2

Saharon Shelah (2001)

Fundamenta Mathematicae

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We continue the work of [2] and prove that for λ successor, a λ-categorical theory T in $L_{κ *, ω}$ is μ-categorical for every μ ≤ λ which is above the $(2^{L S (T)}) ⁺$ -beth cardinal.

CH and the Sacks property

S. Quickert (2002)

Fundamenta Mathematicae

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We show the consistency of CH and the statement “no ccc forcing has the Sacks property” and derive some consequences for ccc $ω^{ω}$ -bounding forcing notions.

Algebraic genericity of strict-order integrability

Luis Bernal-González (2010)

Studia Mathematica

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We provide sharp conditions on a measure μ defined on a measurable space X guaranteeing that the family of functions in the Lebesgue space $L^{p} (μ, X)$ (p ≥ 1) which are not q-integrable for any q > p (or any q < p) contains large subspaces of $L^{p} (μ, X)$ (without zero). This improves recent results due to Aron, García, Muñoz, Palmberg, Pérez, Puglisi and Seoane. It is also shown that many non-q-integrable functions can even be obtained on any nonempty open subset of X, assuming that X is a topological...

Borel classes of uniformizations of sets with large sections

Petr Holický (2010)

Fundamenta Mathematicae

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We give several refinements of known theorems on Borel uniformizations of sets with “large sections”. In particular, we show that a set B ⊂ [0,1] × [0,1] which belongs to $Σ ⁰_{α}$ , α ≥ 2, and which has all “vertical” sections of positive Lebesgue measure, has a $Π ⁰_{α}$ uniformization which is the graph of a $Σ ⁰_{α}$ -measurable mapping. We get a similar result for sets with nonmeager sections. As a corollary we derive an improvement of Srivastava’s theorem on uniformizations for Borel sets with $G_{δ}$ sections. ...

Supercompactness and failures of GCH

Sy-David Friedman, Radek Honzik (2012)

Fundamenta Mathematicae

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Let κ < λ be regular cardinals. We say that an embedding j: V → M with critical point κ is λ-tall if λ < j(κ) and M is closed under κ-sequences in V. Silver showed that GCH can fail at a measurable cardinal κ, starting with κ being κ⁺⁺-supercompact. Later, Woodin improved this result, starting from the optimal hypothesis of a κ⁺⁺-tall measurable cardinal κ. Now more generally, suppose that κ ≤ λ are regular and one wishes the GCH to fail at λ with κ being λ-supercompact. Silver’s...

Fourier analysis, linear programming, and densities of distance avoiding sets in $ℝ^{n}$

Fernando Mário de Oliveira Filho, Frank Vallentin (2010)

Journal of the European Mathematical Society

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We derive new upper bounds for the densities of measurable sets in $ℝ^{n}$ which avoid a finite set of prescribed distances. The new bounds come from the solution of a linear programming problem. We apply this method to obtain new upper bounds for measurable sets which avoid the unit distance in dimensions $2, \dots, 24$ . This gives new lower bounds for the measurable chromatic number in dimensions $3, \dots, 24$ . We apply it to get a short proof of a variant of a recent result of Bukh which in turn generalizes theorems...

The linear refinement number and selection theory

Michał Machura, Saharon Shelah, Boaz Tsaban (2016)

Fundamenta Mathematicae

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The linear refinement number is the minimal cardinality of a centered family in ${[ω]}^{ω}$ such that no linearly ordered set in $({[ω]}^{ω}, \subseteq *)$ refines this family. The linear excluded middle number is a variation of . We show that these numbers estimate the critical cardinalities of a number of selective covering properties. We compare these numbers to the classical combinatorial cardinal characteristics of the continuum. We prove that = = in all models where the continuum is at most ℵ₂, and that the cofinality...

Borel sets with σ-compact sections for nonseparable spaces

Petr Holický (2008)

Fundamenta Mathematicae

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We prove that every (extended) Borel subset E of X × Y, where X is complete metric and Y is Polish, can be covered by countably many extended Borel sets with compact sections if the sections $E_{x} = y \in Y : (x, y) \in E$ , x ∈ X, are σ-compact. This is a nonseparable version of a theorem of Saint Raymond. As a by-product, we get a proof of Saint Raymond’s result which does not use transfinite induction.

On Dimensionsgrad, resolutions, and chainable continua

Michael G. Charalambous, Jerzy Krzempek (2010)

Fundamenta Mathematicae

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For each natural number n ≥ 1 and each pair of ordinals α,β with n ≤ α ≤ β ≤ ω(⁺), where ω(⁺) is the first ordinal of cardinality ⁺, we construct a continuum $S_{n, α, β}$ such that (a) $d i m S_{n, α, β} = n$ ; (b) $t r D g S_{n, α, β} = t r D g o S_{n, α, β} = α$ ; (c) $t r i n d S_{n, α, β} = t r I n d ₀ S_{n, α, β} = β$ ; (d) if β < ω(⁺), then $S_{n, α, β}$ is separable and first countable; (e) if n = 1, then $S_{n, α, β}$ can be made chainable or hereditarily decomposable; (f) if α = β < ω(⁺), then $S_{n, α, β}$ can be made hereditarily indecomposable; (g) if n = 1 and α = β < ω(⁺), then $S_{n, α, β}$ can be made chainable and hereditarily indecomposable. In...

Median for metric spaces

Nacereddine Belili, Henri Heinich (2001)

Applicationes Mathematicae

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We consider a Köthe space $(, | | \cdot | |_{)}$ of random variables (r.v.) defined on the Lebesgue space ([0,1],B,λ). We show that for any sub-σ-algebra ℱ of B and for all r.v.’s X with values in a separable finitely compact metric space (M,d) such that d(X,x) ∈ for all x ∈ M (we then write X ∈ (M)), there exists a median of X given ℱ, i.e., an ℱ-measurable r.v. Y ∈ (M) such that ${| | d (X, Y) | |}_{\leq} | | d (X, Z) | |$ for all ℱ-measurable Z. We develop the basic theory of these medians, we show the convergence of empirical medians and we give...