2018
Том 70
№ 9

All Issues

Banakh T. O.

Articles: 9
Brief Communications (Ukrainian)

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.

Article (English)

Scattered Subsets of Groups

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

↓ Abstract   |   Full text (.pdf)

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 ℵ0 scattered subsets.

Article (English)

On thin-complete ideals of subsets of groups

Banakh T. O., Lyaskovska N.

↓ Abstract   |   Full text (.pdf)

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.

Article (English)

Completeness of invariant ideals in groups

Banakh T. O., Lyaskovska N.

↓ Abstract   |   Full text (.pdf)

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

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

Article (English)

Topological Spaces with Skorokhod Representation Property

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

↓ Abstract   |   Full text (.pdf)

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.

Article (Ukrainian)

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.

↓ Abstract   |   Full text (.pdf)

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)dμ(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.

Brief Communications (Ukrainian)

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

Banakh T. O., Vovk M. I.

↓ Abstract   |   Full text (.pdf)

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 $M ⊂ X × Y$.

Article (Ukrainian)

Symmetric Subsets and Colorings of Connected Compact Groups

Banakh T. O.

↓ Abstract   |   Full text (.pdf)

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.

Article (Ukrainian)

Parametric results for certain infinite-dimensional manifolds

Banakh T. O.

Full text (.pdf)

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