A Lebesgue measurable set in the real line has Lebesgue density 0 or 1 at almost every point. Kolyada showed that there is a positive constant \(\delta\) such that for non-trivial measurable sets there is at least one point with upper and lower densities lying in the interval \((\delta, 1-\delta)\). Both Kolyada and later Szenes gave bounds for the largest possible value of this \(\delta\). In this note we reduce the best known upper bound, disproving a conjecture of Szenes.

Some Schur, Vitali-Hahn-Saks and Nikodým convergence theorems for \((l)\)-group-valued measures are given in the context of \((D)\)-convergence. We consider both the \(\sigma\)-additive and the finitely additive case. Here the notions of strong boundedness, countable additivity and absolute continuity are formulated not necessarily with respect to a same regulator, while the pointwise convergence of the measures is intended relatively to a common \((D)\)-sequence. Among the tools, we use the Fremlin lemma, which allows us to replace a countable family of \((D)\)-sequence with one regulator, and the Maeda-Ogasawara-Vulikh representation theorem for Archimedean lattice groups.

Some properties of Laplace continuous functions and Laplace derivable functions are studied.

We consider a real function \(f\) of a real variable such that, for every point \(x\) of the preimage \(f^{-1}(D)\) of a set \(D \subseteq \mathbb{R}\), \(f\) is strictly monotone at \(x\), and give sufficient conditions of strict monotonicity of \(f\) on \(f^{-1}(D)\). In particular, we prove that a differentiable function \(f\) on an open interval, whose derivative is strictly negative on \(f^{-1}(D)\), where \(D \subseteq \mathbb{R}\) is an open set, is strictly decreasing on \(f^{-1}(D)\). The latter result has applications in stability theory of differential equations on \(\mathbb{R}^N\). The first application provides Lyapunov functions \(V\) for preimages under \(V\) of closed sets. The second application is a generalization of the Lyapunov stability theorem, in which the role of the asymptotically equilibrium point is played by \(V^{-1}(-\infty, c_0]\), where \(V\) is a Lyapunov function for \(V^{-1}(-\infty, c_0]\), and all sublevel sets of \(V\) are assumed to be compact. Moreover, due to compactness, all solutions of the differential equation are global to the right. The second application is also a generalization of a boundedness result from Geophysical Fluid Dynamics; in particular, it proves rigorously that all trajectories of the famous Lorenz system eventually enter a compact set.

This note shows that if a subset \(S\) of \(\mathbb{R}\) is such that some continuous function \(f\colon\mathbb{R}\to\mathbb{R}\) has the property “\(f[S]\) contains a perfect set,” then some \(\mathcal{C}\,^\infty\) function \(g\colon\mathbb{R}\to\mathbb{R}\) has the same property. Moreover, if \(f[S]\) is nowhere dense, then the \(g\) can have the stronger property “\(g[S]\) is perfect.” The last result is used to show that it is consistent with ZFC (the usual axioms of set theory) that for each subset \(S\) of \(\mathbb{R}\) of cardinality \(\mathfrak{c}\) (the cardinality of the continuum) there exists a \(\mathcal{C}\,^\infty\) function \(g\colon \mathbb{R}\to\mathbb{R}\) such that \(g[S]\) contains a perfect set.

Here we present some comparison results between Birkhoff and McShane multivalued integration.

In this paper we describe the structure of the arithmetic sum of two affine Cantor sets. These are self-similar sets which are part of the dynamically defined Cantor sets. Let \({\textbf{C}_1}, {\textbf{C}_2}\) be affine Cantor sets with \([0,s]\) and \([0,r]\) as intervals of step 0. We show a generic family of these self-similar sets for which the structure of \({\textbf{C}_1}+{\textbf{C}_2}\) is of one of the following five types: (i) an \(M\)-Cantorval, (ii) an \(R\)-Cantorval, (iii) an \(L\)-Cantorval, or there exist \(\lambda, \eta >0\) and intervals \(I\), \({\tilde I}\) of the construction of \({\textbf{C}_1}\) and \({\textbf{C}_2}\), respectively, such that (iv) \(\lambda {\textbf{C}_1}+\eta {\textbf{C}_2}= {\textbf{C}_1}\cap I + {\textbf{C}_2}\cap {\tilde I} -\min I - \min {\tilde I} \,= \,[0, \lambda s + \eta r]\),\, or (v) \(\lambda {\textbf{C}_1}+\eta {\textbf{C}_2}= {\textbf{C}_1}\cap I + {\textbf{C}_2}\cap {\tilde I} -\min I - \min {\tilde I}\) is homeomorphic to the Cantor ternary set. This result generalizes the description obtained by Mendes and Oliveira for the case of homogeneous Cantor sets and the one obtained by the first two authors for semi-homogeneous Cantor sets. It also provides a suitable framework in which a question of Mendes and Oliveira admits a positive answer.

We prove that the Hausdorff dimension of the graph of a prevalent continuous function is 2. We also indicate how our results can be extended to the space of continuous functions on \([0,1]^d\) for \(d \in \mathbb{N}\) and use this to obtain results on the ‘horizon problem’ for fractal surfaces. We begin with a survey of previous results on the dimension of a generic continuous function.

We provide a simple construction of a function \(F\colon\mathbb{R}^2\to\mathbb{R}\) discontinuous on a perfect set \(P\), while having continuous restrictions \(F\restriction C\) for all twice differentiable curves \(C\). In particular, \(F\) is separately continuous and linearly continuous. While it has been known that the projection \(\pi[P]\) of any such set \(P\) onto a straight line must be meager, our construction allows \(\pi[P]\) to have arbitrarily large measure. In particular, \(P\) can have arbitrarily large \(1\)-Hausdorff measure, which is the best possible result in this direction, since any such \(P\) has Hausdorff dimension at most 1.

Let \(\mu\) be a Radon probability measure on the Euclidean space \(\mathbb{R}^{d}\) for \(d\geq 1\), and \(f\,:\mathbb{R}^{d}\to \mathbb{R}\) a measurable function. Given a sequence in \((\mathbb{R}^{d})^{\mathbb{N}}\), for any \(x\in\mathbb{R}^{d}\) define \(f_{n}(x)=f(x_{k})\), where \(x_{k}\) is the first among \(x_{0},\ldots, x_{n-1}\) that minimizes the distance from \(x\) to \(x_{k}\), \(0 \leq k\leq n-1\). The measures for which the sequence \((f_{n})_{n=1}^{\infty}\) converges in measure to \(f\) for almost every sequence \((x_{0},x_{1},\ldots)\) are called Mycielski-regular. We show that the self-similar measure generated by a finite family of contracting similitudes and which up to a constant is the Hausdorff measure in its dimension on an invariant set \(C\) is Mycielski-regular.

We compute the dimension spectrum of certain nonconventional averages, namely, the Hausdorff dimension of the set of \(0,1\) sequences, for which the frequency of the pattern 11 in positions \(k, 2k\) equals a given number \(\theta\in [0,1]\).

Some types of compactness in the ideal context are defined and relations between ideal exhaustiveness and equicontinuity of measures are investigated. As applications, some versions of limit theorems involving ideal pointwise convergence of measure sequences and some weak compactness results related to integral functionals are presented.

In this paper, we study the stochastic integral equation with its stochastic integral defined using the Henstock approach, or commonly known as the generalized Riemann approach, instead of the classical Itô integral, which we shall call it the Itô-Henstock integral equation. Our aim is to prove the existence of solution of the Itô-Henstock integral equation using the well known method used in the existence theorem of the ordinary differential equation, namely the Picard’s iteration method.

To every Lebesgue measurable subset of \(\mathbb{R}\) is associated a certain subcollection of points where the given measurable set possesses a density. By virtue of Lebesgue’s famous theorem on metric density, this associated set is a set of full measure in \(\mathbb{R}\) and is hence measure-theoretically very large. But are these sets also topologically large? In Lebesgue’s theorem, the set is kept fixed while the point is allowed to vary. If instead, we keep the point fixed a vary the set, then we may have corresponding to each point in \(\mathbb{R}\) a certain subclass of measurable sets each member of which possesses a density at that point. How large is this subclass in the “topology of measurable subsets of \(\mathbb{R}\)”? In this paper, in an endeavour to seek out answers to the questions set above, we have arrived at certain interesting and significant conclusions. Somewhat similar conclusions have been derived over analogous questions relating to ‘set-porosity’.

Let \(F:[a,b]\longrightarrow \R\) have zero derivative in a dense subset of \([a,b]\). What else we need to conclude that \(F\) is constant in \([a,b]\)? We prove a result in this direction using some new Mean Value Theorems for integrals which are the real core of this paper. These Mean Value Theorems are proven easily and concisely using Lebesgue integration, but we also provide alternative and elementary proofs to some of them which keep inside the scope of the Riemann integral.

The concept of uniform differentiability is introduced to characterize sequences of McShane and Henstock equi-integrable functions.

In this short note, we give a relatively simple proof of the Łoś-Marczewski Extension of finitely additive measures. In particular, we extend the Łoś-Marczewski Extension to Dedekind complete Riesz-space-valued functions.

By assuming the Continuum Hypothesis, it is proved that there exists a subgroup of \(\mathbb{R}^\mathbb{R}\) of cardinality strictly greater than the cardinality of the continuum, all nonzero members of which are absolutely nonmeasurable additive functions.

We give a characterization of the Henstock-Kurzweil integral on \(\R^m\) in terms of variational measures. As an application of this we prove a generalization of the Hake’s theorem to \(\R^m\).

In this short note, we give a simple proof of the Sobczyk-Hammer Decomposition Theorem in terms of Dedekind complete Riesz spaces.

A saturation result with respect to data error level is presented for an approximate derivative operator of Kyrille Popoff.

The notion of \(\mathcal{A}_{d}-\)density point introduced in [Wojdowski 2006] leads to the operator \(\Phi _{\mathcal{A}_{d}}\left( A\right) \) which is not a lower density operator. We present a counterexample and give a corrected definition which should be used in [Wojdowski 2006] and [Wojdowski 2007] to keep all results valid.