We study certain multiple ergodic averages of an iterated functions system generated by two contractions on the unit interval. By using the dynamical coding \(\{0,1\}^{\mathbb{N}}\) of the attractor, we compute the Hausdorff dimension of the set of points with a given frequency of the pattern \(11\) in positions \(k, 2k\).

The purpose of this article is to give a solution to a problem raised by A. Maliszewski in \(\left[5\right]\) by showing that any lower semicontinuous function can be represented as a sum of two lower semicontinuous strong \'{S}wi\k{a}tkowski functions.

We provide a characterization of lower semicontinuity for multifunctions with values in a metric space \(\langle Y,d \rangle\) which, in the special case of single-valued functions, says that a function is continuous if and only if for each \(\varepsilon \gt 0\), the \(\varepsilon\)-tube about its graph is an open set. Applications are given, one of which provides a novel understanding of the Open Mapping Theorem from functional analysis. We also give a related but more complicated characterization of upper semicontinuity for multifunctions with closed values in a metrizable space.

We investigate differences between upper and lower porosity. In finite dimensional Banach spaces every upper porous set is directionally upper porous. We show the situation is very different for lower porous sets; there exists a lower porous set in \(\mathbb{R}^2\) which is not even a countable union of directionally lower porous sets.

In this paper we investigate for which closed subsets \(P\) of the real line \(\mathbb{R}\) there exists a continuous map from \(P\) onto \(P^2\) and, if such a function exists, how smooth can it be. We show that there exists an infinitely many times differentiable function \(f\colon\mathbb{R}\to\mathbb{R}^2\) which maps an unbounded perfect set \(P\) onto \(P^2\). At the same time, no continuously differentiable function \(f\colon\mathbb{R}\to\mathbb{R}^2\) can map a compact perfect set onto its square. Finally, we show that a disconnected compact perfect set \(P\) admits a continuous function from \(P\) onto \(P^2\) if, and only if, \(P\) has uncountably many connected components.

The purpose of this note is to calculate the almost sure Hausdorff dimension of uniformly random self-similar fractals. These random fractals are generated from a finite family of similarities, where the linear parts of the mappings are independent uniformly distributed random variables at each step of iteration. We also prove that the Lebesgue measure of such sets is almost surely positive in some cases.

In the present paper we prove the following theorem: \\ Let \(\{\vf _{m,n}(x,y)\}_{m,n=1}^{\infty} \) be an arbitrary uniformly bounded double orthonormal system on \(I^2:=[0,1]^2\) such that for some increasing sequence of positive integers \(\{N_n\}_{n=1}^\infty \) the Lebesgue functions \(L_{N_n,N_n}(x,y)\) of the system are bounded below a. e. by \( \ln^{1+\epsilon} N_n \), where \(\epsilon \) is a positive constant. Then there exists a function \(g \in L(I^2)\) such that the double Fourier series of \(g\) with respect to the system \(\{\vf _{m,n}(x,y)\}_{m,n=1}^{\infty} \) essentially diverges in measure by squares on \(I^2\). The condition is critical in the logarithmic scale in the class of all such systems %\footnote{ 2000 Mathematics Subject Classification : Primary 42B08; Secondary 40B05. % Key words and phrases: %Essential divergence in measure, orthogonal Fourier series, Lebesgue functions}.

A weak selection on an infinite set \(X\) is a function \(f:[X]^2 \to X\) such that \(f(F) \in F\) for each \(F \in [X]^2 := \{ E \subseteq X : |E| = 2 \}\). If \(f: [X]^2 \to X\) is a weak selection and \(x, y \in \mathbb{R}\), then we say that \(x \lt_f y\) if \(f(\{x, y\}) = x\) and \(x \leq_f y\) if either \(x = y\) or \(x \lt_f y\). Given a weak selection \(f\) on \(X\) and \(x, y \in X\), we let \((x,y]_f = \{z \in X : x \lt_f z \le_f y \}\). If \(f: [\mathbb{R}]^2 \to \mathbb{R}\) is a weak selection and \(A \subseteq \mathbb{R}\), then we define \[ \lambda^{*}_{f}(A):=\inf\Big\{\sum_{n \in \mathbb{N}} |b_{n} - a_{n}| \, : \, A \subseteq \bigcup_{n \in \mathbb{N}}(a_{n},b_{n}]_{f} \Big\} \] if there exists a countable cover by semi open \(f\)-intervals of \(A,\) and if there is not such a cover, then we say that \(\lambda^{*}_{f}(A)=+\infty\). This function \(\lambda^{*}_{f}\:mathcal{P}(\mathbb{R}) \longrightarrow [0,+\infty]\) is an outer measure on the real line \(\mathbb{R}\) which generalizes the Lebesgue outer measure. In this paper, we show several interesting properties of these kind of outer measures.

For families \(\mathcal{F}\) of flats (i.e., affine subspaces) of \(\mathbb{R}^n\), we investigate the classes of \(\mathcal{F}\)-continuous functions \(f\colon\mathbb{R}^n\to\mathbb{R}\), whose restrictions \(f\restriction F\) are continuous for every \(F\in\F\). If \(\mathcal{F}_k\) is the class of all \(k\)-dimensional flats, then \(\mathcal{F}_1\)-continuity is known as linear continuity; if \(\mathcal{F}_k^+\) stands for all \(F\in\mathcal{F}_k\) parallel to vector subspaces spanned by coordinate vectors, then \(\mathcal{F}_1^+\)-continuous maps are the separately continuous functions, that is, those which are continuous in each variable separately. For the classes \(\mathcal{F}=\mathcal{F}_k^+\), we give a full characterization of the collections \(\mathcal{D}\mathcal{F}(\mathcal{F})\) of the sets of points of discontinuity of \(\F\)-continuous functions. We provide the structural results on the families \(\mathcal{D}(\mathcal{F}_k)\) and give a full characterization of the collections \(\mathcal{D}(\mathcal{F}_k)\) in the case when \(k\geq n/2\). In particular, our characterization of the class \(\mathcal{D}(\mathcal{F}_1)\) for \(\mathbb{R}^2\) solves a 60 year old problem of Kronrod.

Some new types of limit theorems for topological group-valued measures are proved in the context of filter convergence for suitable classes of filters. We investigate \((s)\)-boundedness, \(\sigma\)-additivity and regularity properties of topological group-valued measures. We consider also Schur-type theorems, using the sliding hump technique, and prove some convergence theorems in the particular case of positive measures. We deal with the notion of uniform filter exhaustiveness, by means of which we prove some theorems on existence of the limit measure, some other kinds of limit theorems and their equivalence, using known results on existence of countably additive restrictions of strongly bounded measures. Furthermore we pose some open problems.

We define and study, for subsets of \([0, \infty),\) several types of strong right upper porosity at the point 0. Some characterizations of these types of porosity are obtained, including a characterization in terms of a universal property and a characterization in terms of a structural property.

This article presents simple and easy proofs of the Implicit Function Theorem and the Inverse Function Theorem, in this order, both of them on a finite-dimensional Euclidean space, that employ only the Intermediate-Value Theorem and the Mean-Value Theorem. These proofs avoid compactness arguments, the contraction principle, and fixed-point theorems.

The aim of this paper is to give a functional analytic proof of the Lebesgue--Darst decomposition theorem \cite{Darst}. We show that the decomposition of a nonnegative valued additive set function into absolutely continuous and singular parts with respect to another derives from the Riesz orthogonal decomposition theorem employed in a corresponding Hilbert space.

Using different types of absolute continuity, we characterize additive interval functions which are the primitives of Pettis or strongly McShane integrable functions.

After introducing the concepts of \(\varphi\)-derivatives and \(\varphi\)-integrals inside the dual real number algebra, we prove a new generalization of the fundamental theorem of calculus.