Measure Theory in Convergence Spaces Part II: Abstract Integration in Net Convergence Structures

Abstract

The construction of a coherent and general integration procedure is the central question of this second part of our study of measure theory in convergence spaces. We define, and establish convergence theorems for such process of integration, called the net Riemann integration, in convergence groups and commutative monoids, preserving the prescribed convergence throughout. The underlying interchange theorem admits noncommutative groups and requires right completeness only at the cardinality of the error family. Under intrinsic stability assumptions, countable error control therefore needs only sequential completeness, even for arbitrary integration nets, from which we also stablish a general version of Henstock's lemma. The resulting uniform theorems reaches incomplete ranges in which no complete subspace contains a tail of the approximating integrals. In the ordered locally solid convergence theory side, countability of the inner direction replaces countable accessibility of the approximants, and local existence arguments yield monotone and dominated convergence and, under additional controller assumptions, a further extension through increasing integrable barriers with pointwise coverage and uniformly vanishing excess primitives. This constructs the limiting integral without a common integrable envelope and gives uniform control of primitive variation. We finish with two appendices on new and general results of abstract metrization of net convergences and related topics.