Bernoulli sequences and Borel measurability in $(0,1)$

Petr Veselý

Displaying similar documents to “Bernoulli sequences and Borel measurability in $(0, 1)$ ”

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.

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.

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...

Borel spaces

K. P. S. Bhaskara Rao, B. V. Rao

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CONTENTSIntroduction............................................................................... 5Chapter 1. Borel spaces........................................................ 7 § 1. Borel spaces....................................................... 7 § 2. Classical descriptive set theory............................... 10 § 3. Measure and category............................................... 12 § 4. Countably generated structures.............................. 13 § 5. Product...

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...

Pointwise convergence and the Wadge hierarchy

Alessandro Andretta, Alberto Marcone (2001)

Commentationes Mathematicae Universitatis Carolinae

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We show that if $X$ is a $Σ_{1}^{1}$ separable metrizable space which is not $σ$ -compact then $C_{p}^{*} (X)$ , the space of bounded real-valued continuous functions on $X$ with the topology of pointwise convergence, is Borel- $Π_{1}^{1}$ -complete. Assuming projective determinacy we show that if $X$ is projective not $σ$ -compact and $n$ is least such that $X$ is $Σ_{n}^{1}$ then $C_{p} (X)$ , the space of real-valued continuous functions on $X$ with the topology of pointwise convergence, is Borel- $Π_{n}^{1}$ -complete. We also prove a simultaneous improvement of theorems...

On nonmeasurable images

Robert Rałowski, Szymon Żeberski (2010)

Czechoslovak Mathematical Journal

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Let $(X, 𝕀)$ be a Polish ideal space and let $T$ be any set. We show that under some conditions on a relation $R \subseteq T^{2} \times X$ it is possible to find a set $A \subseteq T$ such that $R (A^{2})$ is completely $𝕀$ -nonmeasurable, i.e, it is $𝕀$ -nonmeasurable in every positive Borel set. We also obtain such a set $A \subseteq T$ simultaneously for continuum many relations ${(R_{α})}_{α < 2^{ω}} .$ Our results generalize those from the papers of K. Ciesielski, H. Fejzić, C. Freiling and M. Kysiak.

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...

Displaying similar documents to “Bernoulli sequences and Borel measurability in ( 0 , 1 ) ”

Displaying similar documents to “Bernoulli sequences and Borel measurability in $(0, 1)$ ”