We show that there is no addition preserving Erdős-Sierpiński mapping on any uncountable locally compact abelian Polish group. This generalizes results of Bartoszyński and Kysiak.

We study the notion of \(\psi\)-sparse point and \(\psi\)-sparse topology for nondecreasing continuous function \(\psi\). We show that \(\psi\)-sparse topology is stronger then the \(\psi\)-density topology and weaker than the density topology.

We characterize products of Baire one double star functions.

A solution is presented of a problem proposed at the Summer Symposium in Real Analysis XXXIII.

In this paper we give the definition of \([\lambda,\varrho]\)-continuity of real-valued functions defined on an open interval, which is an example of path continuity. We give some properties of \([\lambda,\varrho]\)-continuous functions. The aim of the paper is to find the maximal additive class and the maximal multiplicative class for the family of \([\lambda,\varrho]\)-continuous functions.

The concepts of uniformly distributed sequences of an increasing family of finite sets and Riemann integrability are considered in terms of the “Lebesgue measure” on infinite-dimensional rectangles in \(R^{\infty}\) and infinite-dimensional versions of famous results of Lebesgue and Weyl are proved.

In the paper there is presented a necessary and sufficient condition to compare \(f\)-density topologies.

New sufficient conditions for representation of a function as an absolutely convergent Fourier integral are obtained in the paper.

We will prove learning theorems that could explain, if only a little, how some organisms generalize information that they get from their senses.

We present a two-dimensional nonabsolute gauge integral which satisfies several convergence theorems and a general divergence theorem, and at the same time admits a change of variables formula valid up to affine transformations - thus applicable to piecewise linear surfaces. Our approach is based on a modification of the $M_1$-integral presented in [6], using triangle-based partitions.

By using the boundedness of the maximal and sharp operators on Morrey spaces, we have proved that the commutators \([M_p,b]\) and \([M^{\#},b]\) are bounded on Morrey spaces \(L^{q,\lambda}\) if and only if \(b\) is in BMO and the negative part of \(b\) is in \(L^{\infty}\).

Let \((X, \mathcal{A}, \mu)\) be a finite measure space, \(E\) a locally convex Hausdorff space, \(L_{E}^{1}\) the space of functions \(f: X \to E\) which are \(\mu\)-integrable by semi-norms, \(P(\mu, E)\) the space of Pettis integrable functions and \(P_{1}(\mu, E)\) those elements of \(P(\mu, E)\) which are measurable by semi-norms. We prove that a linear continuous mapping \( T: L_{E}^{1} \to E\) is of the form \(T(f)= \int g f d \mu\) (\(g \in L^{\infty}\)) if and only if \( h( T(f))=0\) whenever \( h\circ f=0 \) for any \( f \in L_{E}^{1}, h \in E'\). Similar results are proved for \(P(\mu, E)\) and \(P_{1}(\mu, E)\).

We study the compactness of the automorphism group of a domain in \(\mathbb{C}^n\), and in particular the convergence properties of mappings. We supply an application to the semicontinuity of automorphism groups under perturbation of the underlying domain. Relevant examples are provided.

We define a Henstock-type integral for vector valued functions defined in a probability metric compact Radon space, using a suitable family \({\mathcal B}\) of measurable sets which play the role of "intervals". When \({\mathcal B}\) is the family of all subintervals of \([0,1]\) we obtain the classical Henstock-Kurzweil integral on the real line, whereas if \({\mathcal B}\) is the family of all subintervals of \([0,1]^2\), or that of all subintervals of \([0,1]^2\) with a fixed regularity, we obtain the classical Henstock integral on the plane with respect to the Kurzweil base or the Kempisty base respectively.

We investigate continuous piecewise affine interval maps with countably many laps that preserve the Lebesgue measure. In particular, we construct such maps having knot points (a point \(x\) where Dini's derivatives satisfy \(D^{+}f(x)=D^{-}f(x)= \infty\) and \(D_{+}f(x)=D_{-}f(x)= -\infty\)) and estimate their topological entropy. Our main result is: for any \(\varepsilon>0\) we construct a continuous interval map \(g=g_{\varepsilon}\) such that (i) \(g\) preserves the Lebesgue measure; (ii) knot points of \(g\) are dense in \([0,1]\) and for a \(G_{\delta}\) dense set of \(z\)'s, the set \(g^{-1}(\{z\})\) is infinite; (iii) \(h_{\text{top}}(g)\le\log 2+\varepsilon\).

This paper contains the properties of continuous functions equipped with the \(\mathcal{I}(J)\)-density topology or natural topology in the domain or the range.

We prove that for any lower semicontinuous function \(f:[0,1]\to[0,1]\) with purely unrectifiable graph and for any \(\varepsilon>0\) there is an open set \(U\supset \text{graph} f\)\, with every vertical section set of one-dimensional Lebesgue measure at most \(\varepsilon.\)

This note gives another proof of the known fact that \(L^{p}\)-bounded pointwise convergence implies weak convergence in \(L^{p},\) \(p>1.\) The proof is based on Banach and Saks’ theorem. The same method applies to convergence in measure.

We use elementary methods to find pointwise bounds for the first derivative of a real valued function with a continuous, bounded second derivative on an interval.

It was shown in \cite{B} and \cite{Ke} that if \(F(x)\) and \(\Psi(x)\) are Riemann integrals of the form \(\int_a^x f\text{dx}\) and \(\int_b^x \psi \text{dx}\) resp., then \(\psi\cdot f\circ\Psi\), if defined, is Riemann integrable. Furthermore, the change of variable formula applies, giving \(\int_b^x \psi\cdot f\circ\Psi \, \text{dx} = F(\psi(x))-F(\psi(b))\). It is natural to try to generalize this theorem to the Kurzweil-Henstock integral (this question was also dealt with in a paper by the author \cite{Be}); in other words, assuming that \(F\) and \(\Psi\) are KH integrals of \(f\) and \(\psi\) resp., one would expect that \(\psi\cdot f\circ\Psi\) be KH integrable. We show in this paper that this is false, and produce a counterexample based on the middle-third Cantor set and some rudiments of fractal geometry. In other words, by a well known theorem, we prove that the composition of two ACG functions needs not be ACG (in fact, we prove more generally that the composition of two absolutely continuous functions needs not be ACG). Of course, examples that show that the composition of two absolutely continuous functions needs not be absolutely continuous exist in the context of the Lebesgue integral, but since KH integrals need not be absolutely continuous, one cannot infer from these examples the validity of the above claim in the context of KH integration. On the other hand, the subtle method developed in this paper seems to be new, is entirely constructive, and we believe it could be applied to other interesting constructions. \vskip 0pt plus 10pt

In the paper [3] the authors have examined functions of the Baire class 1, where the domain and the range were metric spaces $\varepsilon - \delta$characterization of such functions has been proved. In this note we examine, if replacing of the condition from [3] by its stronger version can lead us to the characterization of some subclass of $B_1$ on the interval $[0,1]$.