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The paper is devoted to construction of some closed inductive sequence of models of the generalized second-order Dedekind theory of real numbers with exponentially increasing powers. These models are not isomorphic whereas all models of the standard second-order Dedekind theory are. The main idea in passing to generalized models is to consider instead of superstructures with the single common set-theoretical equality and the single common set-theoretical belonging superstructures with several generalized equalities and several generalized belongings for rst and second orders. The basic tools for the presented construction are the infraproduct of collection of mathematical systems different from the factorized Los ultraproduct and the corresponding generalized infrafiltration theorem. As its auxiliary corollary we obtain the generalized compactness theorem for the generalized second-order language.

The remarkable class of measurable functions was introduced by classics of function theory. It has found different applications in various branches of mathematics. However this class turned out too restrictive for solving some natural mathematical problems because it is essentially connected with the property of countability. Therefore, along with it another remarkable class, essentially connected with the property of finiteness, was introduced. It is the class of uniform functions. Measurable functions are described both in the classical Lebesgue–Borel language of preimages and in the quite new language of covers. Uniform functions are described in the language of covers exclusively. Both the families of measurable functions and the families of uniform functions are determined by the rigid structure of their supports (descriptive spaces). For this reason, mathematicians weakened more that once the rigidity of the structure of descriptive spaces at the expense of using the additional property of negligence. The present paper is devoted to a contemporary formalization of the indicated ideas. Some applications of the introduced classes of functions to solving a number of known mathematical problems is traced in the paper.

The classical works in function theory (by E. Borel, R. Baire, H. Lebesgue, F. Hausdorff, et al.) laid down the foundation of the classical descriptive theory of functions. Its initial notions are those of a descriptive space and of a measurable function on it. Measurable functions are defined in the (classical) pre-image language. However, a specific range of tasks in the theory of functions, measure theory, and integration theory that emerge on this basis necessitates using an entirely different (postclassical) cover language, which is equivalent to the preimage one in the classical case. By means of the cover language, the general concepts of a prescriptive space and of distributable and uniform functions on it are introduced and their basic properties are studied.

It is well known that the family of all continuous functions on a topological space contains all constant functions and is closed with respect to the usual pointwise operations (addition, multiplication, finite supremum and infimum, division) and uniform convergence. The complete description of such function families (normal families) was given by Borel, Lebesgue, and Hausdorff. The normal families turned out to be exactly the families of all functions measurable with respect to multiplicative σ-additive families of sets. In 1914, Hausdorff also described the normal envelope of an arbitrary family of functions. If uniform convergence is replaced by pointwise convergence, then the notions of completely normal family and completely normal envelope arise. In 1977, Regoli described all completely normal families. They turned out to be precisely the families of all functions measurable with respect to σ-algebras of sets. Moreover, Regoli described the completely normal envelope of a specific family of functions. The present paper gives descriptive and some constructive characterizations of the completely normal envelope of an arbitrary family of functions.

There are two widely known functional hierarchies on a topological space \((T,\mathcal {G})\). The transfinite chain \(A(T)\subset\dots\subset \operatorname{Lim}_{\alpha}A(T) \subset\dots\subset \operatorname{Lim}_{\omega_{1}}A(T)\), where A(T) is an initial family of functions on T and \(\operatorname{Lim}_{\alpha}A(T)\) consists of all pointwise limits of sequences of functions from preceding classes, is called the Baire convergence hierarchy. The transfinite chain \(M(T,\mathcal {B}_{0})\subset\dots\subset M(T,\mathcal {B}_{\alpha})\subset\dots\subset M(T,\mathcal {B}(T,\mathcal {G}))\), where \(\mathcal {B}_{0}\subset \dots \mathcal {B}_{\alpha}\subset \dots \subset \mathcal {B}(T,\mathcal {G})\) is some hierarchy in the σ-algebra \(\mathcal {B}(T,\mathcal {G})\) of Borel sets and \(M(T,\mathcal {B}_{\alpha})\) is the family of all \(\mathcal {B}_{\alpha}\)-measurable functions, is called the Borel descriptive hierarchy.

The problem of characterization of integrals as linear functionals is considered in the paper. It starts from the familiar results of F. Riesz (1909) and J. Radon (1913) on integral representation of bounded linear functionals by Riemann–Stieltjes integrals on a segment and by Lebesgue integrals on a compact in \( {\mathbb{R}^n} \), respectively. After works of J. Radon, M. Frechet, and F. Hausdorff the problem of characterization of integrals as linear functionals took the particular form of the problem of extension of Radon’s theorem from \( {\mathbb{R}^n} \) to more general topological spaces with Radon measures. This problem has turned out difficult and its solution has a long and rich history. Therefore, it may be naturally called the Riesz–Radon–Frechet problem of characterization of integrals. The important stages of its solution are connected with such mathematicians as S. Banach, S. Saks, S. Kakutani, P. Halmos, E. Hewitt, R. E. Edwards, N. Bourbaki, V. K. Zakharov, A. V. Mikhalev, et al. In this paper, the Riesz–Radon–Fr´echet problem is solved for the general case of arbitrary Radon measures on Hausdorff spaces. The solution is given in the form of a general parametric theorem in terms of a new notion of the boundedness index of a functional. The theorem implies as particular cases well-known results of the indicated authors characterizing Radon integrals for various classes of Radon measures and topological spaces.

Borel, Lebesgue, and Hausdorff described all uniformly closed families of real-valued functions on a set T whose algebraic properties are just like those of the set of all continuous functions with respect to some open topology on T. These families turn out to be exactly the families of all functions measurable with respect to some σ-additive and multiplicative ensembles on T. The problem of describing all uniformly closed families of bounded functions whose algebraic properties are just like those of the set of all continuous bounded functions remained unsolved. In the paper, a solution of this problem is given with the help of a new class of functions that are uniform with respect to some multiplicative families of finite coverings on T. It is proved that the class of uniform functions differs from the class of measurable functions.

For Borel functions on a perfect normal space and a perfect topological space there are two Baire convergence classifications: one due to Lebesgue and Hausdorff and the other due to Banach. However, neither classification is valid for an arbitrary topological space. In this paper the Baire convergence classification of Borel functions on an arbitrary space is given. This classification of Borel functions uses two classifications of Borel sets: one generalises the Young-Hausdorff classification for a perfect space and the other is new.Bibliography: 17 titles.

We study expansions of functions in the space with respect to systems similar to orthogonal ones. We find estimates for the coefficients and sufficient conditions on them under which the corresponding expansions converge in . These results are analogues of the well-known Hausdorff-Young-Riesz and Hardy-Littlewood-Paley theorems in the theory of trigonometric and orthogonal series. It is shown that the resulting estimates are more exact than the classical ones even in the case of orthogonal systems.

The classics of function theory (E. Borel, H. Lebesgue, R. Baire, W. H. Young, F. Hausdorff, et al.) have laid down the foundation of the classical descriptive theory of functions. Its initial notions are the notions of a descriptive space and of a measurable function on it. Measurable functions were defined in the classical preimage language. However, a specific range of tasks in theory of functions, measure theory, and integration theory emergent on this base necessitates the usage of the entirely different postclassical cover language, equivalent to the preimage language in the classical case. By means of the cover language, the general notions of a prescriptive space and distributable and uniform functions on it are introduced in this paper and their basic properties are studied.