The article surveys some selected topics in descriptive set theory, showing how the problem of classifying sets according to their complexity has been tackled since Cantor's creation of the subject to current research.

The regular open subsets of a topological space form a Boolean algebra, where the join of two regular open sets is the interior of the closure of their union. A content is a finitely additive measure on this Boolean algebra, or on one of its subalgebras. We develop a theory of integration for such contents. We then explain the relationship between contents, residual charges, and Borel measures. We show that a content can be represented by a normal Borel measure, augmented with a liminal structure, which specifies how two or more regular open sets share the measure of their common boundary. In particular, a content on a locally compact Hausdorff space can be represented by a normal Borel measure and a liminal structure on the Stone-Čech compactification of that space. We also show how contents can be represented by Borel measures on the Stone space of the underlying Boolean algebra of regular open sets.

A cliquish function $f: \mathbb R\to\mathbb R$ is internally cliquish if its set of discontinuity points is nowhere dense. A Baire $1$ function $f: \mathbb R\to\mathbb R$ is internally Baire $1$ if its set of discontinuity points is nowhere dense. In this paper we proved that the set of Darboux internally cliquish (Baire $1$) functions is dense and $\sigma$-strongly porous in the family of Darboux cliquish (Baire $1$) functions.

We study contact 3-manifolds using the techniques of sub-Riemannian geometry and geometric measure theory, in particular establishing properties of their Lipschitz homotopy groups. We prove a biLipschitz version of the Theorem of Darboux: a contact $(2n+1)$-manifold endowed with a sub-Riemannian structure is locally biLipschitz equivalent to the Heisenberg group $\mathbb H^n$ with its Carnot-Carathéodory metric. Then each contact $(2n+1)$-manifold endowed with a sub-Riemannian structure is purely $k$-unrectifiable for $k>n$. We then extend results of Dejarnette et al. [5] and Wenger and Young [20] on the Lipschitz homotopy groups of $\mathbb H^1$ to an arbitrary contact 3-manifold endowed with a Carnot-Carathéodory metric, namely that for any contact 3-manifold the first Lipschitz homotopy group is uncountably generated and all higher Lipschitz homotopy groups are trivial. Therefore, in the sense of Lipschitz homotopy groups, a contact 3-manifold is a $K(\pi,1)$-space with an uncountably generated first homotopy group. Along the way, we prove that each open distributional embedding between purely 2-unrectifiable sub-Riemannian manifolds induces an injective map on the associated first Lipschitz homotopy groups. Therefore, each open subset of a contact 3-manifold determines an uncountable subgroup of the first Lipschitz homotopy group of the contact 3-manifold.

This paper has three main results. The first result is an extension theorem for charges on the Boolean algebra of regular open sets of a topological space. The second result is an abstract form of the first result, which we call an abstract extension theorem for charges on Boolean algebras. The third result is a characterization of regular open sets in $R$. The first two results are useful for integration of functions on topological spaces with respect to charges on Boolean algebras of regular open sets. The third result is useful in proving that the Lebesgue measure is not a charge when restricted to the Boolean algebra of regular open sets (see [3])

We give an elementary framework of generalized convexity in terms of comparison of a given function with reference functions from a chosen set. As important examples we consider reference sets used in Alexandrov geometry and give simple proofs for the characterization of generalized convexity in form of differential inequalities.

This note aims to present some rudimentary aspects of relatively bounded nonlinear (not necessarily linear) operators. To be precise, we prove that the set of all nonlinear operators between two normed linear spaces that are relatively bounded with respect to a fixed operator between these spaces itself is a normed linear space under a suitable norm. Furthermore, it is shown that if the codomain is a Banach space, then the space of all relatively bounded nonlinear operators is also complete. Some elementary results on the algebraic structure of the space of all nonlinear operators that are bounded with respect to a fixed operator are also recorded. The results reported in this note illustrate that with some predictable minor modifications, the fundamental classical results on the space of all bounded linear operators can be carried over to a more general setting. Here the word `general' is used in a double sense that the boundedness is generalized to relative boundedness and the restriction of linearity is dropped to include nonlinear maps.

We show that for any closed set $E\subset \lbrack 0,1]$, there exists $f:[0,1]\rightarrow \lbrack 0,1]$ in the first class of Baire that generates $E$ as one of its $\omega $-limit sets. Since $\omega $-limit sets are necessarily closed, this characterizes the class of $\omega $-limit sets for Baire or Borel functions on the interval.

Motivated from the study of multiple ergodic average, the investigation of multiplicative shift spaces has drawn much of interest among researchers. This paper focuses on the relation of topologically mixing properties between multiplicative shift spaces and traditional shift spaces. Suppose that $\mathsf{X}_\Omega^{(\ell)}$ is the multiplicative subshift derived from the shift space $\Omega$ with given $\ell > 1$. We show that $\mathsf{X}_\Omega^{(\ell)}$ is (topologically) transitive/mixing if and only if $\Omega$ is extensible/mixing. After introducing $\ell$-directional mixing property, we derive the equivalence between $\ell$-directional mixing property of $\mathsf{X}_\Omega^{(\ell)}$ and weakly mixing property of $\Omega$.

In this note, we prove a characterization of reflexive Banach spaces with unconditional basis and study the set theoretical complexity of certain sets associated to linear bounded operators acting on reflexive separable Banach spaces.

In this article, we investigate set theoretic operators preserving the ideal Hausdorff convergence of sequences of sets. Then we showed that pointwise continuous functions are not sufficient to preserve ideal Hausdorff convergence. Finally, we proved that uniformly continuous functions preserve this type of convergence.

I prove a mass transference principle for general shapes, similar to a recent result by H. Koivusalo and M. Rams. The proof relies on Vitali's covering lemma and manipulations with Riesz energies.

The main novelty is that it is proved that the obtained limsup-set belongs to the classes of sets with large intersections, as defined by K. Falconer. This has previously not been proved for as general shapes as in this paper.

In this paper we study some properties concerning the second order differentiability of the functions belonging to the Takagi class as well as the size of the sets where these properties hold. In particular, we characterize the set of points where these functions have a Taylor expansion of order two. Moreover, we also characterize when they satisfy a Stepanov condition of order two at a point. In this context, the convexity of such functions is also investigated. Finally, some interesting examples are provided.

We define a sequence of functions that converges uniformly to the decreasing rearrangement $f^*$ of a continuous function $f$ defined on [0,1).

We address a few corrections to [1], mainly concerning Theorem 4 and Corollary 5. Most notably, part (iii) of Theorem 4 should be omitted and the statement of theorem modified slightly. The proofs are identical except for a few typographical corrections.