On some conditions which imply the continuity of almost all sections
Let be an open interval, a topological space and a metric space. Some local conditions implying continuity and quasicontinuity of almost all sections of a function are shown.
On some dimension problems for self-affine fractals.
On some extensions of -finite measures.
On some ideals and related algebras of sets in the plane
On some numerical characterization of Boolean algebras
On some problem of A. Rosłanowski
We present a negative answer to problem 3.7(b) posed on page 193 of [2], where, in fact, A. Rosłanowski asked: Does every set of Lebesgue measure zero belong to some Mycielski ideal?
On some properties of Hamel bases
On some properties of Hamel bases and their applications to Marczewski measurable functions
We introduce new properties of Hamel bases. We show that it is consistent with ZFC that such Hamel bases exist. Under the assumption that there exists a Hamel basis with one of these properties we construct a discontinuous and additive function that is Marczewski measurable. Moreover, we show that such a function can additionally have the intermediate value property (and even be an extendable function). Finally, we examine sums and limits of such functions.
On some properties of Hausdorff content related to instability.
On some properties of Hurewicz, Menger, and Rothberger
On Some Properties of Separately Increasing Functions from [0,1]ⁿ into a Banach Space
We say that a function f from [0,1] to a Banach space X is increasing with respect to E ⊂ X* if x* ∘ f is increasing for every x* ∈ E. A function is separately increasing if it is increasing in each variable separately. We show that if X is a Banach space that does not contain any isomorphic copy of c₀ or such that X* is separable, then for every separately increasing function with respect to any norming subset there exists a separately increasing function such that the sets of points of discontinuity...
On some properties of sets with positive measure
On some properties of submeasures on MV-algebras
On some properties of the Cantor set
On some properties of the Cantor set and the construction of a class of sets with Cantor set properties
On some properties of transformations of a logic
On spaces of measurable functions
On s-sets in spaces of homogeneous type
Let (X,d,μ) be a space of homogeneous type. We study the relationship between two types of s-sets: relative to a distance and relative to a measure. We find a condition on a closed subset F of X under which F is an s-set relative to the measure μ if and only if F is an s-set relative to δ. Here δ denotes the quasi-distance defined by Macías and Segovia such that (X,δ,μ) is a normal space. In order to prove this result, we prove a covering type lemma and a type of Hausdorff measure based criterion...
On strong differentiability of integrals along different directions.