This is a survey of several approaches to the framework for working with infinitesimals and infinite numbers, originally developed by Abraham Robinson in the 1960s, and their constructive engagement with the Cantor-Dedekind postulate and the Intended Interpretation hypothesis. We highlight some applications including (1) Loeb’s approach to the Lebesgue measure, (2) a radically elementary approach to the vibrating string, (3) true infinitesimal differential geometry. We explore the relation of Robinson’s and related frameworks to the multiverse view as developed by Hamkins.

A Besicovitch set is a subset of \(\mathbb{R}^d\) that contains a unit line segment in every direction and the famous Kakeya conjecture states that Besicovitch sets should have full dimension. We provide a number of results in support of this conjecture in a variety of contexts. Our proofs are simple and aim to give an intuitive feel for the problem. For example, we give a very simple proof that the packing and lower box-counting dimension of any Besicovitch set is at least \((d+1)/2\) (better estimates are available in the literature). We also study the “generic validity” of the Kakeya conjecture in the setting of Baire Category and prove that typical Besicovitch sets have full upper box-counting dimension. We also study a weaker version of the Kakeya problem where unit line segments are replaced by half-infinite lines. We prove that such “half-extended Besicovitch sets” have full Assouad dimension. This can be viewed as full resolution of a (much weakened) version of the Kakeya problem.

We consider the generalized Egorov’s statement (Egorov’s Theorem without the assumption on measurability of the functions, see Weiss (2004)) in the case of an ideal convergence and a number of different types of ideal convergence notions. We prove that in those cases the generalized Egorov’s statement is independent from ZFC.

We introduce the notion of the McShane integral in the limit for functions defined on a \(\sigma\)-finite outer regular quasi Radon measure space \((S,\Sigma,\mathcal{T},\mu)\) into a Banach space \(X\) and we study its relation with the generalized McShane integral introduced by D. H. Fremlin. It is shown that if a function from \(S\) into \(X\) is McShane integrable in the limit on \(S\) and scalarly locally \(\tau\)-upper McShane bounded for some \(\tau >0\), then it is McShane integrable on \(S\). On the other hand, we prove that if an \(X\)-valued function is McShane integrable in the limit on \(S\), then it is McShane integrable on each member of an increasing sequence \((S_\ell)_{\ell\geq 1}\) of measurable sets of finite measure with union \(S\). We also prove a version of Beppo Levi’s theorem for this new integral.

We prove several theorems concerning the exceptional sets of the Hilbert transform on the real line. In particular, it is proved that any null set is an exceptional set for the Hilbert transform of an indicator function. The paper also provides a real variable approach to the Kahane-Katsnelson theorem on divergence of Fourier series.

In this article, we make a comprehensive study of the properties of multiplication operators acting on the spaces of functions of bounded \(p\)-variation in Wiener’s sense, \(WBV_p[0,1]\). We characterize all functions \(u\in WBV_p[0,1]\) that define invertible, compact and Fredholm multiplication operators \(M_u\) on \(WBV_p[0,1]\). Also we characterize when \(M_u\) has finite range and has closed range on \(WBV_p[0,1]\).

The Carathéodory theorem on the construction of a measure is generalized by replacing the outer measure with an approximation of it and generalizing the Carathéodory measurability. The new theorem is applied to obtain dynamically defined measures from constructions of outer measure approximations resulting from sequences of measurement pairs consisting of refining \(\sigma\)-algebras and measures on them which need not be consistent. A particular case when the measurement pairs are given by the action of an invertible map on an initial \(\sigma\)-algebra and a measure on it is also considered.

Can a positive function on \(\mathbb{R}\) have zero Lebesgue integral? It depends on how much choice one has.

In the present paper, we extend Gamidov’s integral inequalities to time scales. The obtained results can be used as tools in the study of certain properties of dynamical equations on time scales.

The analogy between measure and Baire category is displayed first by a theorem of Steinhaus and its “dual,” a theorem of Piccard. These two theorems are then applied to provide a double criterion for the unconditional convergence of a series in terms of the “measure size” and the “category size“ of the set of its convergent subseries. As a further application, after a substantial preparatory section concerning essential separability of measurable and \(BP\)-measurable functions, the results about exhaustivity of \(BP_r\)-measurable and universally measurable additive maps on the Cantor group are established. In the last sections of the paper, two classical theorems about countable additivity of the universal measurable and \(BP_r\)-measurable additive maps are examined. The analogy in question is illustrated not only by the results themselves, but also by the proofs provided.

A simple proof of a well-known monotonicity theorem for the approximate derivative is presented. The same technique provides a proof of a generalized version of Ornesteinʼs Theorem.