Recent zbMATH articles in MSC 28A05https://www.zbmath.org/atom/cc/28A052021-04-16T16:22:00+00:00WerkzeugA domain-theoretic investigation of posets of sub-\(\sigma\)-algebras (extended abstract).https://www.zbmath.org/1456.060112021-04-16T16:22:00+00:00"Battenfeld, Ingo"https://www.zbmath.org/authors/?q=ai:battenfeld.ingoSummary: Given a measurable space \((X,\mathcal{M})\) there is a (Galois) connection between sub-\(\sigma\)-algebras of \(\mathcal{M}\) and equivalence relations on \(X\). On the other hand equivalence relations on \(X\) are closely related to congruences on stochastic relations. In recent work, Doberkat has examined lattice properties of posets of congruences on a stochastic relation and motivated a domain-theoretic investigation of these ordered sets. Here we show that the posets of sub-\(\sigma\)-algebras of a measurable space do not enjoy desired domain-theoretic properties and that our counterexamples can be applied to the set of smooth equivalence relations on an analytic space, thus giving a rather unsatisfactory answer to Doberkat's question.
For the entire collection see [Zbl 1391.03010].A. Kharazishvili's some results of on the structure of pathological functions.https://www.zbmath.org/1456.260032021-04-16T16:22:00+00:00"Kirtadze, Aleks"https://www.zbmath.org/authors/?q=ai:kirtadze.aleks-p"Pantsulaia, Gogi"https://www.zbmath.org/authors/?q=ai:pantsulaia.gogi-rauliThe authors present a brief survey of A. Kharazishvili's works devoted to real-valued functions with strange, pathological and paradoxical structural properties, e.g. absolutely non-measurable functions, SierpiĆski-Zygmund functions, sup-measurable and weakly sup-measurable functions of two real variables, and non-measurable functions of two real variables for which both iterated integrals exist.
Reviewer: George Stoica (Saint John)The strong uniqueness property of invariant measures in infinite dimensional topological vector spaces.https://www.zbmath.org/1456.280012021-04-16T16:22:00+00:00"Khachidze, Marika"https://www.zbmath.org/authors/?q=ai:khachidze.marika"Kirtadze, Aleks"https://www.zbmath.org/authors/?q=ai:kirtadze.aleks-pIn [\textit{A. B. Kharazishvili}, Bull. Acad. Sci. GSSR 114, No. 1, 41--48 (1984)] a normalized \(\sigma\)-finite metrically transitive Borel measure \(\chi\) in \(\mathbb{R}^\omega\) is constructed which is invariant with respect to
\[G=c_{00}:=\{x \in \mathbb{R}^\omega : \ \hbox{supp}~ x < \omega\}\]
Let \(s_0\) be the central symmetry of \(\mathbb{R}^\omega\) with respect to the origin and let \(S_\omega\) be the group generated by \(s_0\) and \(G\). In the present paper a \(\sigma\)-finite Borel measure on \(\mathbb{R}^\omega\) is costructed which is invariant with respect to the group \(S_\omega\).
Reviewer: Daniele Puglisi (Catania)