Improper Integrals of Type II—A New Look and Some Surprising Facts

The goal of this work is to present a new theoretical and simultaneously practical approach to the conception of improper integrals of type II and, consequently, to improper integrals of type I. Seven distinct definitions of improper integrals of type II are presented, each illustrated by suitable examples. Among them, particularly important are those defined on nets composed of sets symmetric with respect to points belonging to a given finite family of points. Connections between improper integrals of type II defined here are described in two main theorems. It should be emphasized that all definitions of improper integrals presented here are special cases of a very general definition of improper integrals on non-compact sets, which is presented at the end of this work. This definition is inspired by a concept introduced by Maurin.