Absolutely Integrable Functions

Abstract

Summary The goal of this article is to clarify the relationship between Riemann’s improper integrals and Lebesgue integrals. In previous articles [6], [7], we treated Riemann’s improper integrals [1], [11] and [4] on arbitrary intervals. Therefore, in this article, we will continue to clarify the relationship between improper integrals and Lebesgue integrals [8], using the Mizar [3], [2] formalism.

Integral of Continuous Functions of Two Variables

“…Proof: For every real number e such that 0 < e there exists a real number r such that 0 < r and for every points z 1 , z 2 of (the real normed space of R) × (the real normed space of R) such that z 1 , z 2 ∈ E and z 1 − z 2 < r holds f /z 1 − f /z 2 < e. (11) Let us consider intervals I, J. Then (i) I × J is a subset of (the real normed space of R) × (the real normed space of R), and…”

“…So far, the authors have proved in Mizar [2], [15] many theorems on the integral theory of one-variable functions for Riemann and Lebesgue integrals [9], [5], [11] (for interesting survey of formalizations of real analysis in another proof-assistants like ACL2 [13], Isabelle/HOL [12], Coq [3], see [4]). As a result, we have shown that if a function bounded on a closed interval (i.e., a continuous function) is Riemann integrable, then it is Lebesgue integrable, and both integrals coincide [10].…”

Multidimensional Measure Space and Integration

“…In this paper, using the Mizar system [1], [11], we introduce multidimensional measure spaces and the integration ( [14], [2]) of functions on these spaces (for interesting survey of formalizations of real analysis in another proof-assistants like ACL2 [10], Isabelle/HOL [9], Coq [3], see [4]). It is the continuation of the mechanisation of this topic as developed in [5] and [8]. In constructing measures on multidimensional spaces [12], we constructed a finite sequence of Cartesian product spaces of sets in Section 1.…”