# Banakh T. O.

### Descriptive complexity of the sizes of subsets of groups

Banakh T. O., Protasov I. V., Protasova K. D.

↓ Abstract

Ukr. Mat. Zh. - 2017. - 69, № 9. - pp. 1280-1283

We study the Borel complexity of some basic families of subsets of a countable group (large, small, thin, rarefied, etc.) determined by the sizes of their elements. The obtained results are applied to the Czech – Stone compactification $\beta G$ of the group $G$. In particular, it is shown that the closure of the minimal ideal $\beta G$ has the $F_{\sigma \delta}$ type.

### Scattered Subsets of Groups

Banakh T. O., Protasov I. V., Slobodianiuk S. V.

Ukr. Mat. Zh. - 2015. - 67, № 3. - pp. 304-312

We define scattered subsets of a group as asymptotic counterparts of the scattered subspaces of a topological space and prove that a subset *A* of a group *G* is scattered if and only if *A* does not contain any piecewise shifted *IP* -subsets. For an amenable group *G* and a scattered subspace *A* of *G,* we show that *μ*(*A*) = 0 for each left invariant Banach measure *μ* on *G.* It is also shown that every infinite group can be split into ℵ_{0} scattered subsets.

### On thin-complete ideals of subsets of groups

Ukr. Mat. Zh. - 2011. - 63, № 6. - pp. 741-754

Let $F \subset \mathcal{P}_G$ be a left-invariant lower family of subsets of a group $G$. A subset $A \subset G$ is called $\mathcal{F}$-thin if
$xA \bigcap yA \in \mathcal{F}$ for any distinct elements $x, y \in G$. The family of all $\mathcal{F}$-thin subsets of G is denoted by $\tau(\mathcal{F})$.
If $\tau(\mathcal{F}) = \mathcal{F}$, then $\mathcal{F}$ is called thin-complete.
The *thin-completion* $\tau*(\mathcal{F})$ of $\mathcal{F}$ is the smallest thin-complete subfamily of $\mathcal{P}_G$ that contains $\mathcal{F}$.
Answering questions of Lutsenko and Protasov, we prove that a set $A \subset G$ belongs to $\tau*(G)$ if and only if for any sequence $(g_n)_{n\in \omega}$ of non-zero elements of G there is $n\in \omega$ such that
$$\bigcap_{i_0,...,i_n \in \{0, 1\}}g_0^{i_0}...g_n^{i_n} A \in \mathcal{F}.$$
Also we prove that for an additive family $\mathcal{F} \subset \mathcal{P}_G$ its thin-completion $\tau*(\mathcal{F})$ is additive. If the group $G$ is countable and torsion-free, then the completion $\tau*(\mathcal{F}_G)$ of the ideal $\mathcal{F}_G$ of finite subsets of $G$ is coanalytic and not Borel in the power-set $\mathcal{P}_G$ endowed with the natural compact metrizable topology.

### Completeness of invariant ideals in groups

Ukr. Mat. Zh. - 2010. - 62, № 8. - pp. 1022–1031

We introduce and study various notions of completeness of translation-invariant ideals in groups.

### Topological Spaces with Skorokhod Representation Property

Banakh T. O., Bogachev V. I., Kolesnikov A. V.

Ukr. Mat. Zh. - 2005. - 57, № 9. - pp. 1171–1186

We give a survey of recent results that generalize and develop a classical theorem of Skorokhod on representation of weakly convergent sequences of probability measures by almost everywhere convergent sequences of mappings.

### Direct and Inverse Problems of Baire Classification of Integrals Depending on a Parameter

Banakh T. O., Kutsak S. M., Maslyuchenko O. V., Maslyuchenko V. K.

Ukr. Mat. Zh. - 2004. - 56, № 11. - pp. 1443-1457

We study the problem of the Baire classification of integrals *g* (*y*) = (*If*)(*y*) = ∫ _{X} *f*(*x, y*)*d*μ(*x*), where *y* is a parameter that belongs to a topological space *Y* and *f* are separately continuous functions or functions similar to them. For a given function *g*, we consider the inverse problem of constructing a function *f* such that *g* = *If*. In particular, for compact spaces *X* and *Y* and a finite Borel measure μ on *X*, we prove the following result: In order that there exist a separately continuous function *f* : *X* × *Y* → ℝ such that *g* = *If*, it is necessary and sufficient that all restrictions *g*|_{ Y } _{ n } of the function *g*: *Y* → ℝ be continuous for some closed covering { *Y* _{ n } *: n* ∈ ℕ} of the space *Y*.

### Separately $Fσ$-measurable functions are close to functions of the first baire class

Ukr. Mat. Zh. - 2004. - 56, № 4. - pp. 573–576

We prove that a Borel separately $Fσ$-measurable function $f: X \times Y → R$ on the product of Polish spaces is a function of the first Baire class on the complement $X × Y \backslash M$ of a certain projectively meager set $M ⊂ X × Y$.

### Symmetric Subsets and Colorings of Connected Compact Groups

Ukr. Mat. Zh. - 2001. - 53, № 5. - pp. 694-697

We find upper and lower bounds for the Haar measure of a monochromatic symmetric subset, which can be found in every measurable *r*-coloring of a connected compact group.

### On universality of countable powers of absolute retracts

Ukr. Mat. Zh. - 1996. - 48, № 4. - pp. 540-542

We construct an absolute retract *X* of arbitrarily high Borel complexity such that the countable power *X* ^{ω} is not universal for the Borelian class A _{1} of sigma-compact spaces, and the product *X* ^{ω} x ∑, where ∑ is the radial interior of the Hilbert cube, is not universal for the Borelian class A _{2} of absolute *G* _{δσ}-spaces.

### Parametric results for certain infinite-dimensional manifolds

Ukr. Mat. Zh. - 1991. - 43, № 6. - pp. 853–859