This is a survey paper on some classical trigonometric families of thin sets (Dirichlet sets, weak Dirichlet sets, N-sets, N₀-sets, A-sets, U-sets, and two recently introduced families of B-sets and of B₀-sets), the relationships between them, and basic closure properties of these families, presented as complete answers to ten questions. However, a large part of the paper is devoted to presentation of new results. In addition, we tried to give an overview of the best known estimates for cardinal characteristics for these families and for the families of particularly "permitted" sets, using small uncountable cardinals recently studied in infinite combinatorics. Almost all results are accompanied by brief notes on the investigations preceding them. Finally, we study properties of families of thin sets related to the Rademacher and Walsh orthogonal systems of functions. Some of these families are studied for the first time.

This survey article gives a history of the development of the theory concerning restriction theorems in real analysis. The discussion will span the history from Lusin's Theorem through very recent results concerning intersections of Lipschitz, smooth, and Holder class functions.

The main goal of this paper is to generalize several results concerning cardinal invariants to the statements about the associated families of sets. We also discuss the relationship between the additive properties of sets and their Borel images. Finally, we present estimates for the size of the smallest set which is not strongly meager.

Suppose that $Ω$ is a set, $F$ is a field of subsets of $Ω$, and $A(ℝ)(F)$ is the set of all real-valued finitely additive functions defined on $F$. For each $ξ ϵ A(ℝ)(F)$ let $A(ξ) = \{η : η ϵ A(ℝ)(F), ξ – η \mathrm{bounded}\}$. It is shown that for each $ξ in A(ℝ)(F)$, for certain subsets of $A(ξ)$ with special closure properties, theorems about closest approximations, functional equations and decompositions hold.

A number of invariant extensions of the Lebesgue measure are constructed. In connection with such extensions the class of negligible sets and the class of absolutely negligible sets are investigated. Also the Steinhaus property for these extensions is considered.

In this paper we investigate the Darboux right absorption property of a family of function. In particular, the considerations concentrate around the maximal class of functions having the Darboux right absorption property.

A recent theorem of the author for continuous functions of bounded variation is extended to include the discontinuous case. Given $h$ of bounded variation on a closed interval $K$ let $s(t, y)$ be the total number 0, 1, 2 of the following conditions which hold at $(t, y) ϵ K × ℝ$:

It is investigated the maximal families (additive, multiplicative, lattice and with respect to the composition) for the class of strongly quasicontinuous functions.

It is well known that the following theorem due to Banach and Zarecki: $AC = VB ⋂ (N) ⋂ \mathcal{C}$, on a closed set. In [1] we showed that this theorem is no longer true if $AC$ and $VB$ are replaced by Foran's conditions $A(2)$ and $B(2)$, respectively. In the present paper, we introduce the classes $AC_∞$ and $VB_∞$, which contain strictly the classes $AC$ and $VB$, respectively. Then we show that $AC_∞ = VB_∞ ⋂ (N)$, for bounded measurable functions on a measurable set.

It is shown that for every $Λ$-sequence $Λ$ there is an equivalent $Λ$-sequence $Γ = (ϒ_i)$ such that lim sup $ϒ_{i+1}/ϒ_i = 1$.

It is shown that if $f : ℝ → ℝ$ has a finite $\mathcal I$-approximate derivative ${f'_{{\mathcal I} - ap}}$ everywhere in $ℝ$, then there is a sequence of perfect sets, $H_n$, whose union is $ℝ$, and a sequence of differentiable functions, $h_n$, such that $h_n = f$ over $H_n$ and ${{h'}_n} = {{f'}_{{\mathcal I} - ap}}$ over $H_n$. This result is a complete analogue of that on approximately differentiable functions by R. J. O'Malley.

Let $\mathcal A$ stand for the class of all almost continuous functions from $ℝ$ to $ℝ$ and let $\mathrm A(\mathcal A)$ be the smallest cardinality of a family $F ⊆ ℝ^ℝ$ for which there is no $g: ℝ → ℝ$ with the property that $f + g ϵ \mathcal A$ for all $f ϵ F$. We define cardinal number $A(\mathcal D)$ for the class $\mathcal D$ of all real functions with the Darboux property similarly. It is known, that $c < \mathrm A(\mathcal A) ≤ 2^c$ [11]. We will generalize this result by showing that the cofinality of $\mathrm A(\mathcal A)$ is greater that $c$. Moreover, we will show that it is pretty much all that can be said about $\mathrm A(\mathcal A)$ in ZFC, by showing that $\mathrm A(\mathcal A)$ can be equal to any regular cardinal between $c⁺$ and $2^c$ and that it can be equal to $2^c$ independently of the cofinality of $2^c$. This solves a problem of T. Natkaniec [11, Problem 6.1, p. 495].

We will also show that $\mathrm A(\mathcal D) = \mathrm A(\mathcal A)$ and give a combinatorial characterization of this number. This solves another problem of Natkaniec. (Private communication.)

In this article I prove a few theorems concerning representations of functions in a given family of functions with a common bound as the sums of bounded Darboux functions with a common summand, which are analogous to the results of Fast, Mišík and Pu & Pu.

In this paper, we show that the projective tensor product of a two-dimensional $\ell ^p$ space with a two-dimensional $\ell ^q$ space never has the Mazur Intersection Property for a large range of values of $p$ and $q$. For this purpose, we characterize the extreme contractions from $\ell _2^p$ to $\ell _2^q$ and obtain their closure.

In 1977 Leindler and Németh [4] proved a general theorem concerning integrability of a power series. The present note deads with a similar problem for Dirichlet series.

In this short note, we give a more 'direct' proof of a classical theorem of Dunford, Pettis and Phillips, that any weakly compact operator from $L^1(μ)$ into a Banach space $X$ is representable.

Several properties (measurability, quasicontinuity and cliquishness) of functions of two variables having special sections are proved.

Various types of locad convergence of sequences of functions are investigated in this paper. This investigation is based on the concept of oscillation of sequence of function. Applications of proved results to differentiability of real functions are given.

We prove that Peano derivatives and approximate Peano derivatives of measurable functions are measurable.

A locally Lipschitz function is regular at points where its lower Dini and Clarke derivatives coincide. For a locally Lipschitz function on an open interval the set of points where the function is regular but not differentiable is at most countable. By constructing a set with unusual metric density properties, and integrating its characteristic function, we produce a locally Lipschitz function which is nowhere regular.

A measure space $(X, S, μ)$ is called almost finite if $X$ is a union of a set of finite measure and finitely many atoms of infinite measure. It is shown that Egoroff's Theorem for sequences of measurable functions holds if and only if the underlying measure space is almost finite. As a consequence we obtain several theorems on the interaction between convergences almost everywhere, almost uniform and in measure, respectively, with no preliminary conditions on the measure space $(X, S, μ)$, thus extending results from [2], [4], [6] and [10]. It is proved further that if $(X, S, μ)$ is almost finite (is not almost finite), then $ɸ : ℝ → ℝ$ preserves almost uniform convergence and convergence in measure, respectively, if and only if $ϕ$ is continuous (is uniformly continuous), thus augmenting a result of [3].

If $F$ is a continuous function of intervals in $ℝ^m$, then its distribution function is continuous. The converse is true if $m = 1$ but false if $m ≥ 2$. In the present note we prove these facts and we explain why the onedimensional case is an exception.

We prove some results concerning covering, and translation properties of Hamel bases. For example, we show that the complement of a union of fewer than continuum many translates of a Hamel basis, more generally Erdős set, is everywhere of the second category.

A local version of the relative continuity of J. Chew and J. Tong is introduced and observed to have applications to connected functions and separate continuity. A strong version of Levine's decomposition of continuity is also obtained.

The Khintchine integral is not comparable to the approximately continuous Perron integral. That is, there are functions that are integrable in one sense but not the other. There have been several attempts to define an integration process that includes both of these integrals. This paper points out an error in one of these proofs and defines another integration process that includes both of the integrals mentioned above.

We give necessary and sufficient conditions for a function to be general Denjoy or Kubota integrable. The conditions we obtain are in terms of a sequence of Lebesgue integrable functions converging to the given function.