Related papers: Haar-$\mathcal I$ sets: looking at small sets in P…

We show that under some conditions on a family $\mathcal{A}\subset\bbi$ there exists a subfamily $\mathcal{A}_0\subset\mathcal{A}$ such that $\bigcup \mathcal{A}_0$ is nonmeasurable with respect to a fixed ideal $\bbi$ with Borel base of a…

We show that the enveloping space $X_G$ of a partial action of a Polish group $G$ on a Polish space $X$ is a standard Borel space, that is to say, there is a topology $\tau$ on $X_G$ such that $(X_G, \tau)$ is Polish and the quotient Borel…

In this paper we introduce the notion of $\mathcal I^{\mathcal K}$-Cauchy function, where $\mathcal I$ and $\mathcal K$ are ideals on the same set. The $\mathcal I^{\mathcal K}$-Cauchy functions are a generalization of $\mathcal I^*$-Cauchy…

We extend \L ukasiewicz logic obtaining the infinitary logic $\mathcal{IR}\L$ whose models are algebras $C(X,[0,1])$, where $X$ is a basically disconnected compact Hausdorff space. Equivalently, our models are unit intervals in…

Several recent articles in operator algebras make a nontrivial use of the theory of measurable fields of von Neumann algebras $(M_x)_{x \in X}$ and related structures. This includes the associated field $(\text{Aut}\ M_x)_{x \in X}$ of…

We introduce the concept of an $\mathrm{L}^{1}$ full group associated with a measure-preserving action of a Polish normed group on a standard probability space. These groups carry a natural Polish group topology induced by an $\mathrm{L}^1$…

In the paper we will show how to construct a Haar meager set (consequently meager) which is not Haar null, and conversely, a meager Haar null set which is not Haar meager in spaces of sequences: $l_p$ with $p\geq1$, $c_0$ or $c$. It refers…

Denote by $U_{\mathcal I}({\mathcal H})$ the group of all unitary operators in ${\bf 1}+{\mathcal I}$ where ${\mathcal H}$ is a separable infinite-dimensional complex Hilbert space and ${\mathcal I}$ is any two-sided ideal of ${\mathcal…

We consider the isometry group of the infinite dimensional separable hyperbolic space with its Polish topology. This topology is given by the pointwise convergence. For non-locally compact Polish groups, some striking phenomena like…

\emph{Minkowski rings} are certain rings of simple functions on the Euclidean space $W = {\mathbb{R}}^d$ with multiplicative structure derived from Minkowski addition of convex polytopes. When the ring is (finitely) generated by a set…

A $\sigma$-ideal $\mathcal{I}$ on a Polish group $(X,+)$ has Smital Property if for every dense set $D$ and a Borel $\mathcal{I}$-positive set $B$ the algebraic sum $D+B$ is a complement of a set from $\mathcal{I}$. We consider several…

Let $G$ be a connected semi-simple group defined over and algebraically closed field, $T$ a fixed Cartan, $B$ a fixed Borel containing $T$, $S$ a set of simple reflections associated to the simple positive roots corresponding to $(T,B)$,…

We study several notions of null sets on infinite-dimensional Carnot groups. We prove that a set is Aronszajn null if and only if it is null with respect to measures that are convolutions of absolutely continuous (CAC) measures on Carnot…

The symmetric inverse semigroup $I(X)$ on a set $X$ is the collection of all partial bijections between subsets of $X$ with composition as the algebraic operation. We study a minimal Hausdorff inverse semigroup topologies on $I(X)$. When…

The Wadge hierarchy was originally defined and studied only in the Baire space (and some other zero-dimensional spaces). We extend it here to arbitrary topological spaces by providing a set-theoretic definition of all its levels. We show…

We study the Borel subsets of the plane that can be made closed by refining the Polish topology on the real line. These sets are called potentially closed. We first compare Borel subsets of the plane using products of continuous functions.…

Suppose $X$ is a locally compact Polish space, and $G$ is a group of lattice isometries of $C_0(X)$ which satisfies certain conditions. Then we can equip $C_0(X)$ with an equivalent lattice norm $| \! | \! | \cdot | \! | \! |$ so that $G$…

Given a class C of subgroups of a topological group G, we say that a subgroup H in C is a universal C subgroup of G if every subgroup K in C is a continuous homomorphic preimage of H. Such subgroups may be regarded as complete members of C…

L\'evai and Pyber proposed the following as a conjecture: Let $G$ be a profinite group such that the set of solutions of the equation $x^n=1$ has positive Haar measure. Then $G$ has an open subgroup $H$ and an element $t$ such that all…

Lecture notes in Russian. Topics: the Haar measure (abstract theorems and explicit descriptions for different groups), measures on infinite-dimensional spaces with large natural groups of symmetries (Gaussian measures, Poisson measures,…