The genesis of this paper goes back to a question posed years ago by the late H. Steinhaus, “About two plane arcs both of finite lengths it is known that every line in the plane meets both at the same number of points (which may be zero or $\infty$). Must the arcs be identical ?” This question in answered in this paper using the Crofton transform. For a fixed set in the plane the Crofton transform for that set is defined as the number of points in which the set meets a variable line in the plane. In this paper we construct an inversion of the Crofton transform within a certain class of plane Borel sets, a somewhat weak form of inversion retrieving from the Crofton transform (actually from the cross-integral function, which is somewhat stronger) the closure of the set rather than the set (or the set modulo a set of linear measure zero) itself. We also establish uniqueness (or rather a degree of uniqueness) of two plane sets from the same class, whose Crofton transforms coincide over certain families of lines. This answers a stronger version of the original Steinhaus question.

The Radon transform is considered as a mapping from Euclidean n-space to a radially weighted mixed norm space. This leads to a strong analog of the Hausdorff-Young theorem for dimensions $n\ge 3$. A similar estimate is obtained for the Radon transform over the complex domain which holds for all dimensions $n\ge 2$.

The paper constructs a division space from an arbitrary non-atomic measure space with a locally compact topology that is compatible with the measure, and defines two equivalent integrals.

In a recent paper “Limits without epsilons” by Darwin E. Peek, the concept of the convergence of real sequences has been completely characterized by six properties of a relation on sequences (determining Cauchy sequences). With the exception of the third property, called the “Squeezing Theorem”, the characterization is shown to be minimal in the sense that each of the other five conditions is necessary to preserve the characterization. We answer in the positive the question whether also the remaining property is necessary and hence independent of the other five. Our construction is based on the properties of vector ${\mathcal{L}}_0^{*}$-groups.

In this paper it is proved that each a.e. continuous Baire one function $u:{ℝ}^m\to ℝ$ can be written as $u=f\cdot g+h$, where $f$, $g$ and $h$ are a.e. continuous derivatives (with respect to the ordinary differentiation basis).

We give an example of a measurable set $E\subseteq ℝ$ such that the set $E\prime =\{(x,y):x+y\in E\}$ is not in the $\sigma$-algebra generated by the rectangles with measurable sides. We also prove a stronger result that there exists an analytic (${\sum }_1^1$) set $E$ such that $E^{\prime }$ is not in the $\sigma$-algebra generated by rectangles whose horizontal side is measurable and vertical side is arbitrary. The same results are true when measurable is replaced with property of Baire.

In this article we calculate the exact Borel class of important collection of thin sets from harmonic analysis, the closed $H$-sets in the unit circle. We show that the $H$-sets are properly $G_{\delta \sigma }$.

An example is given of a function $f:ℝ\to ℝ$ which satisfies the property that for each $x$ there is an such that $\left|f(x+h)-f(x-h)\right|\ge 1$ whenever . Further properties of such functions are also discussed.

Recently, Freiling and Humke answered a question of the present author concerning the density property of certain sets. In this paper we examine how their work affects two different but related theorems. The first theorem concerns the monotonicity of approximately semicontinuous functions and the second, the existence of approximate maxima of approximately continuous functions.

In this paper we investigate properties of Carathéodory’s superposition and show some of their applications to differential equations.

Let $I$ be a non-degenerate interval in ${ℝ}^n$; $J$, $B$ and $L$ are the Jordan, Borel and Lebesgue sets respectively in $I$; $\sigma (J)$ is the $\sigma$-algebra generated by $J$; $F_0=\{n\hspace{0.17em}:\hspace{0.17em}n\text{ is a subset of an }{F}_{\sigma }\text{ subset of }I\text{ having Lebesgue measure zero}\}$; $S_0=\{b\cup n\hspace{0.17em}:\hspace{0.17em}b\in B,n\in F_0\}$; $S_1=\{b\cup n:b\in B,n\text{ is a first category subset of }I\text{ having Lebesgue measure zero}}$. The symbol “$\subset$” denotes “proper subset”.

THEOREM 1. $\sigma (J)=S_0$.

THEOREM 2. $B\subset \sigma (J)\subset S_1$.

The purpose of this paper is to study a behavior of continuous metric preserving functions $f$ with $f^{\prime }(0)=+\infty$. First we show, via a simple example, that it is possible that such a function has no finite derivatives at any point. Then in Example 2 we construct a nondecreasing, differentiable, metric preserving function having infinite derivative at least at the points $x=2^{-n}$ for each natural number, $n$.

This paper is a continuation of the authors’ earlier paper [4]. Denote by $\mathcal{M}$ the metric space of all metrics on a given nonvoid set $X$ with the sup-metric. In this paper some subsets of $\mathcal{M}$ are investigated, namely subsets consisting of all metrics $d\in \mathcal{M}$ for which $(X,d)$ possesses prescribed topological (metric) properties.

We show that in the space, $\mathcal{D}{B}_1$, of Darboux Baire $1$ functions on the unit interval, equipped with the metric of uniform convergence, the subspace $\mathcal{G}$, consisting of those functions whose graph restricted to the points of continuity is dense in the entire graph, is closed and nowhere dense, and that the subspace, $\mathcal{D}{B}_1^{*}$, of Darboux Baire* $1$ functions is first category relative to $\mathcal{G}$; or stated in an equivalent manner, the space of universally first return continuous functions is a closed nowhere dense subspace of the space of first return continuous functions, and that $\mathcal{D}{B}_1^{*}$ is first category in the space of universally first return continuous functions.

Two different notions of global porosity are discussed. Relations between them are discovered, and some new characterizations of globally porous sets are given. A connection to Dolženko’s boundary value problem is emphasized. Finally the notion of a bilaterally globally porous set is defined and some properties of such sets are discused. Several errors in the literature are corrected.