Classification of the measurable functions of several arguments and invariant measures on the space of matrices and tensors, 1. Classificaton. Content 1.Introduction: list of the problems. 2.Classificaton of measurable functions:one variable and pure functions. 2.1.Classical case-functions of one variable. 2.2.Pure functions. 3.Complete metric invariant of the measurable functions of several ar...

This paper focuses on fuzzy measurable functions based on gradual numbers. Firstly, some new concepts about gradual numbers are introduced as preparation. And then, the concept of gradual numbervalued measurable functions is introduced and some of its properties are investigated. Mathematics Subject Classification: 03E72, 28E10

Based on the concept of an interval-valued function which is motivated by the goal to represent an uncertain function, we define the Choquet integral with respect to a fuzzy measure of intervalvalued functions. We also discuss convergence in the C mean and convergence in a fuzzy measure of sequences of measurable interval-valued functions. In particular, we investigate the convergence theorem f...

For a long time people have been trying to develop probability theory starting from ‘finite’ events rather than collections of infinite events. In this way one can find natural replacements for measurable sets and integrable functions, but measurable functions seemed to be more difficult. We present a solution. Moreover, our results are constructive (in the sense of Bishop).

Let (Ω, M, µ) be a measure space. Then for any two complex valued measurable functions f and g, we have the Cauchy-Schwarz inequality: 0.1 THEOREM (Cauchy-Schwarz inequality). Let (Ω, M, µ) be a measure space. Then for any two complex valued measurable functions f and g such that

Based on a minimal set of axioms we introduce a general integral which can be defined on arbitrary measurable spaces. It acts on measures which are only (finite) monotone set functions and on measurable functions whose range is contained in the unit interval. We introduce the notion of integral equivalence of pairs of measures and functions which leads us to a special important general integral...