2018
Том 70
№ 12

# Banakh T. O.

Articles: 9
Brief Communications (Ukrainian)

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

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

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

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

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

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

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

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

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

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