Let \((X,\rho)\) be a complete separable metric space and \(\mathcal M\) be the set of all probability Borel measures on \(X\). We show that if the space \(\mathcal M\) is equipped with the weak topology, the set of measures having the upper (resp. lower) correlation dimension zero is re\-si\-dual. Moreover, the upper correlation dimension of a typical (in the sense of Baire category) measure is estimated by means of the local lower entropy and local upper entropy dimensions of \(X\).

In this paper a type of path derivative, which is not based on the non-empty path intersection, is introduced. Such derivatives share some basic properties with sequencial derivatives but there exist some sharp contrasts between them. The path derivative here restricts the speed of convergence in the definition of limits naturally and has a sharp contrast with the classical Dini derivatives for the typical continuous function.

Let \(X\) be a separable Banach space, \(Y\) a Banach space and \(f:X \to Y\) a Lipschitz function. We show that the set of all G\^ ateaux non-differentiability points at which \(f\) has all one-sided or two-sided directional derivatives can be covered by (special subsets of) Lipschitz surfaces of codimension 1 or codimension 2, respectively. Further results indicate that these results are close to the best possible ones. Our results are new also for Lipschitz functions \(\R^n \to \R\); for these functions G\^ ateaux differentiability is the classical (total) differentiability.

Let \(0<a<\sqrt 2\). Suppose \(\delta=\delta(d,\varepsilon)\) has the following property. If \(\mathcal N\) is an \(a\)-net of the Euclidean ball in \(\mathbb{R}^{d}\), \(A\subset \mathcal N\), and \(f:A\to \mathbb{R}^d\) is \((1+\varepsilon)\)-bilipschitz, then \(f\) admits a \((1+\delta)\)-bilipschitz extension \(f:\mathcal N\to \mathbb{R}^d\). We give some estimates of \(\delta\).

It is known that any subspace \(\H\) of the space of continuous functions on a compact set can be represented as the space of affine continuous functions defined on the state space of \(\H\). The aim of this paper is to generalize this result for abstract affine functions of various descriptive classes (Borel, Baire etc.). The important step in the proof is to derive results on the preservation of the descriptive properties of topological spaces under perfect mappings. The main results are applied on the space of affine functions on compact convex sets and on approximation of semicontinuous and Baire--one abstract affine functions.

We examine for which sets \(H\) of the circle group \(\mathbb{R} / \mathbb{Z}\) can the difference functions \(f(x+h)- f(x)\) of a measurable or \(L_{p}\) function \(f\) belong to an \(L_{q}\) class for every \(h \in H\) without \(f\) itself being in \(L_{q}\). Tamás Keleti conjectured in \cite{Elek1} that these sets are the \(N\)-sets; that is, the sets of absolute convergence of Fourier-series. We prove this conjecture for \(q \leq 2\). For \(q=2\), as a quantitative analogue of this statement, we prove a minimax theorem.

In this article we construct an example of a bilaterally quasicontinuous Darboux function \(f:[0,1] \to [0,1]\), which has no fixed points.

Let \((\Omega,\mu)\) be a finite measure space, \(\Phi(t)=\int_{0}^{t} a(s)\, ds\) and \(\Psi(t)=\int_{0}^{t} b(s)\, ds\), where \(a\) and \(b\) are positive continuous functions defined on \([0,\infty)\). Consider the associated Orlicz spaces \(L^{\Phi}(\Omega)\) and \(L^{\Psi}(\Omega)\). In this paper we find a relationship between \(a\) and \(b\) to assure that \(T\), a sublinear and positive homogeneous operator of restricted weak type \((p,p)\) and of type \((\infty,\infty)\), maps \(L^{\Psi}(\Omega)\) into \(L^{\Phi}(\Omega)\). If the two Orlicz spaces are normable, our results imply the continuity of \(T\). This relation between \(a\) and \(b\) is sharp since it is shown to be necessary for operators like the one side maximal operators related to the Ces\`aro averages.

For subharmonic functions \(u\geq 0\) in the unit ball \(B_N\) of \(\mathbb{R}^N \), the paper characterizes this kind of growth: \(\sup_{x\in B_N} (1- \vert x\vert ^2 )^\alpha u(x) <+\infty \) (given \(\alpha >0\)), through criteria involving such integrals as \(\int u(x)\, dx \) or \(\int u(x) ( 1- \vert x\vert ^2 )^{\alpha -N} \, dx \) over balls centered at points \(a\in B_N\). Given \(p \in \mathbb{R}\) and \(\omega\) some non--negative function, this article compares subharmonic functions with the previous kind of growth to subharmonic functions satisfying: \( \sup_{a\in B_N} \int_{ B_N } u(x) ( 1- \vert x\vert ^2 )^p \omega (\vert \varphi _a (x)\vert)\, dx <+\infty \), where \(\varphi _a\) are Möbius transformations. The paper also studies subharmonic functions which are sums of lacunary series and their links with both previous kinds of subharmonic functions.

Let \(\mathcal{A}\) be an algebra of subsets of an underlying set \(A\) (which is not the entire power set of \(A\) in general), and let \({\mathcal I} \subseteq {\mathcal A}\) be an ideal over \(A\). The pair \(({\mathcal A}, {\mathcal I})\) is said to have the {\em hull property} iff whenever \(X \subseteq A\), there is a \(Y \in {\mathcal A}\) such that \(X \subseteq Y\) and \(Y\) is ``least'' mod \({\mathcal I}\), i.e., if \(Z \in {\mathcal A}\) and \(X \subseteq Z\), then \(Y \setminus Z \in {\mathcal I}\). It has been observed that in many cases for which \(({\mathcal A}, {\mathcal I})\) satisfies the hull property, the quotient Boolean algebra \({\mathcal A}/{\mathcal I}\) is a complete Boolean algebra. That, and the superficial similarity between the definitions themselves, along with the similar proofs that have sometimes resulted when using these properties, leads to the natural question of how the two properties ``\(({\mathcal A}, {\mathcal I})\) satisfies the hull property'' and ``\({\mathcal A}/{\mathcal I}\) is a complete Boolean algebra'' are related to each other. Examples will be produced which show that neither of these two properties implies the other. In addition, we examine the question of what additional hypotheses would cause one of these properties to imply the other.

In this paper we introduce the class of extra strong \'Swi\c{a}tkowski functions, and we characterize the following maximal classes for the families of strong \'Swi\c{a}tkowski and extra strong \'Swi\c{a}tkowski functions: the maximal additive class, the maximal multiplicative class, and the maximal class with respect to maximums.

It is proved that if \(B\) is a convex quasi-density basis and \(E\) is a symmetric space on \(\mathbb{R}^n\) with respect to Lebesgue measure, then there do not exist non-orthogonal weights \(w\) and \(v\) for which the maximal operator \(M_B\) corresponding to \(B\) acts compactly from the weight space \(E_w\) to the weight space \(E_v\).

We show that for every finite set \(0\notin S\subset \mathbb{Z}^{d}\) with the property \(-S=S\), every real trigonometric polynomial \(f\) on the \(d\) dimensional torus \(\mathbb{T}^{d}=\mathbb{R}^{d}/\mathbb{Z}^{d}\) with spectrum in \(S\) has a zero in every closed ball of diameter \(D\left(S\right)\), where \(D\left(S\right)=\sum _{\lambda \in S}\frac{1}{4||\lambda ||_{2}}\), and investigate tightness in some special cases.

A set in a separable metric space is called \emph{Borel--opaque} if it meets every Borel set of positive topological dimension. We show that if there is a set of reals with cardinality of the continuum and universal measure zero, then each separable space contains a Borel--opaque set that is of universal measure zero. Similar results hold for opaque sets that are perfectly meager, \la sets, \lap sets etc., and can be extended to some nonseparable spaces. On the other hand, we show that a \s set is zero--dimensional. Using opacity we also construct universal measure zero sets of positive Hausdorff dimension.

For a symmetrically continuous function \(f : \mathbb R\to [0,1]\), a reduction formula is obtained which gives a Darboux symmetrically continuous function \(g_f : \mathbb R\to [0,1]\) such that the set \(C(f)\) of continuity points of \(f\) is a subset of \(C(g_f)\). Under additional conditions, \(g_f\) and the oscillation function \(\omega_f\) of \(f\) are Croft-like functions. One consequence of \(g_f\) being Darboux is that the absolutely convergent values \(s(x)\) of a real trigonometric series \(\sum_{n=1}^{\infty} \rho_n \sin(nx + x_n)\), with \(\sum_{n=1}^{\infty} | \rho_n |=\infty\) and with an uncountable set \(E\) of points of absolute convergence, almost has the intermediate value property except for countably many values \(s(x)\) and countably many points of \(E\).

It is shown that Martin's Axiom for \(\sigma\)-centered partial orders implies that every maximal orthogonal family in \(\mathbb{R}^\mathbb{N}\) is of size \(2^{\aleph_0}\).

Let \(T\) be an infinite set and \(\mathcal{I}\) be a fixed ideal on \(T\). We introduce and study the notion of almost constant convergence of sequences \(\{ f_t\colon t\in T\}\) of real functions with respect to the ideal \(\mathcal{I}\). This notion generalizes discrete convergence, transfinite convergence, and \(\omega_2\)-convergence. In particular, we consider the question when a given family of functions (e.g., continuous, Baire class 1, Borel measurable, Lebesgue measurable, or functions with the Baire property) is closed with respect to this kind of convergence.

We investigate the algebraic sums of sets for a large class of invariant \(\sigma\)-ideals and \(\sigma\)-fields of subsets of the real line. We give a simple example of two Borel subsets of the real line such that its algebraic sum is not a Borel set. Next we show a similar result to Proposition 2 from A. Kharazishvili paper \cite{S2}. Our results are obtained for ideals with coanalytical bases.

In this paper we show that the multifractal Hausdorff measure and multifractal packing measure introduced by Olsen and Peyriere can be expressed as Henstock-Thomson \lq\lq variation" measures. As an application we prove a density theorem for these two measures that extends results by Edgar and is more refined than those found in [Ol1].

General notions of integration have been introduced by Saks \cite[p. 254]{S3}, Kart\'ak \cite[p. 482]{K14}, Kubota \cite[p. 389]{K8} and Sarkhel \cite[p. 299]{S19}. Kart\'ak's \(T\)-integration was further studied by Kart\'ak and Ma\v{r}\`{\i} in \cite{K15}, and by Kubota in \cite{K17}. \par In this paper, starting from Kartak and Ma\v{r}\`{\i}'s definition, we introduce another general integration (see Definition \ref{D3}), that allows a very general theorem of dominated convergence (see Theorem \ref{T2}). Then we present a general definition for primitives, and this definition contains many of the known nonabsolutely convergent integrals: the Denjoy\(^*\)-integral, the \(\alpha\)-Ridder integral, the wide Denjoy integral, the \(\beta\)-Ridder integral, the Foran integral, the AF integral, the Gordon integral. Using this integration and Theorem \ref{T2}, we obtain a generalization of a result on differential equations, of Bullen and Vyborny \cite{B17}. \par We further give a Banach-Steinhaus type theorem, a categoricity theorem, Riesz type theorems (as a particular case we obtain the Alexiewicz Theorem \cite{A2}), and study the weak convergence for the \(T\)-integration.

A theorem is presented which shows the uniform closure of the class of uniformly polygonally approximable functions cannot be distinguished from the class of delta-fine functions by the nature of their exceptional sets of quasicontinuity.

The \(GR_k\) integral, a Stieltjes type integral, was introduced by the authors. In this paper we introduce a \(GR_k\)-type integral that admits the fundamental theorem of calculus. Also, a convergence theorem is proved using equi-integrability conditions.

In this paper we introduce the idea of a function having the strong \'Swi\c{a}tkowski property at a point. The main result of this work is proving that the class of functions which have the %``locally" strong \'Swi\c{a}tkowski property at each point, is equal to the class of all strong \'Swi\c{a}tkowski functions.

A function \(f:\mathbb R \to \mathbb R\) satisfies the condition \(({\mathcal A}_3)\) if for each real \(r > 0\), for each \(x\) and for each set \(U \ni x\) belonging to the density topology there is an open interval \(I\) such that \(C(f) \supset I \cap U \neq \emptyset \) and \(f(U\cap I) \subset (f(x)-r,f(x)+r)\), where \(C(f)\) denotes the set of all continuity points of \(f\). In this article we investigate the sets \(A\) such that each almost continuous function may be extended from \(A\) to a function having property \(({\mathcal A}_3)\).

In this paper, a controlled convergence theorem is proved for \(n\)-dimensional strong variational Banach-valued integrals, also referred herein as Banach-valued Multiple Integrals. The methods used in the proof for one dimensional case given in [15], in which linearization was used, cannot be applied for the higher dimensional case. Instead, we follow the ideas in [17, Chapter 5, Section 21; 4; 18].

We show that the class of (dyadic) wavelet sets is in one-to-one correspondence to a special class of Lebesgue measurable isomorphisms of \([0,1)\) which we call {\it wavelet induced} maps. We then define two natural classes of maps \(\fwi\) and \(\swi\) which, in order to simplify their construction, retain only part of the characterization properties of a wavelet induced map. We prove that each wavelet induced map appears from the Schröder-Cantor-Bernstein construction applied to some \(u\in \mathcal{WI}_1\) and \(v\in \mathcal{WI}_2\). Consequently, the construction of a wavelet set is basically equivalent to the easier construction of two maps \(u\in \mathcal{WI}_1\) and \(v\in \mathcal{WI}_2\). Some older results on wavelet sets are recovered using this new point of view. The connectivity result of Speegle (\cite{sp}) is recaptured and the completeness in the natural metric of the class of wavelet sets is reestablished. Although these ideas seem to generalize to more than one dimension, specific examples are given only in the one dimensional case.

The purpose of this article is to prove Hahn Decomposition type and Jordan Decomposition type theorems for measures on \(\sigma\)-semirings. These results generalize the classical theorems for measures on \(\sigma\)-algebras.

We find an easy way to construct a continuous nowhere differentiable function from any nondecreasing convex function mapping the unit interval onto itself. We give a number of examples of nowhere differentiable functions constructed this way.

In paper \cite{Mar} it is proved that every jump function \(f:{\mathbb R} \to {\mathbb R}\) is the sum of two unilaterally continuous jump functions. In this article we prove that the analogous result is not true for the density topology. Moreover we show some necessary and sufficient conditions ensuring that an approximate jump function is the sum of two unilaterally approximately continuous and approximate jump functions.

In the space \(B_1(0,1])\) of all Baire 1 real functions on \([0,1]\), equipped with the topology of pointwise convergence there is an uncountable discrete closed subset of Darboux functions. This affirmatively answers a question of A. Leiderman.

This paper contains two results related to the question of when a measure on a metric space is determined by its value on certain subsets. The first is that two finite positive measures on a countable metric abelian group \(G\) which agree on all balls of some fixed non-zero radius agree on \(G\). The second relates to measures on a compact metric space that agree on all intersections of pairs of balls.

We consider real-valued functions defined on the unit interval. It is known that the class of first-return recoverable functions is the same as the class of polygonally approximable functions and that this class consists of the Baire one functions. Here we introduce the more restrictive classes of consistently first-return recoverable functions and consistently polygonally approximable functions. We show these classes are identical and consist of those functions which are continuous except at countably many points.

A well known result proved by F. John for \(0<\la<1/2\) and by J. O. Strömberg for \(\lambda=1/2\) states that \[\|f\|_{BMO(\o)}\asymp\sup_Q\inf_{c\in {\mathbb R}} \inf\{\a>0: \o\{x\in Q:|f(x)-c|>\a\}<\la\o(Q)\}\] for any measure \(\omega\) satisfying the doubling condition. In this note we extend this result to all absolutely continuous measures. In particular, we show that Str\"omberg's ``1/2-phenomenon" still holds in the nondoubling case. An important role in our analysis is played by a weighted rearrangement inequality, relating any measurable function and its John-Str\"omberg maximal function. This inequality was proved earlier by the author in the doubling case; here we show that actually it holds for all weights. Also we refine a result due to B. Jawerth and A. Torchinsky, concerning pointwise estimates for the John-Str\"omberg maximal function.