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Abstract
Basic limit theorems for the KH integral involve equiintegrable sets. We construct a family of Banach spaces X^sub Δ^ whose bounded sets are precisely the subsets of KH[0, 1] that are equiintegrable and pointwise bounded. The resulting inductive limit topology on U^sub Δ^ X^sub Δ^ = KH[0, 1] is barreled, bornological, and stronger than both pointwise convergence and the topology given by the Alexiewicz seminorm, but it lacks the countability and compatibility conditions that are often associated with inductive limits.
Key Words: barreled, convergence rate, Denjoy space, equiintegrable, Henstock integral, inductive limit, KH integral, Kurzweil integral, norm
Mathematical Reviews subject classification: 26A39
Received by the editors May 12, 2004
Communicated by: Stefan Schwabik
(ProQuest Information and Learning: ... denotes formulae omitted.)
1 Introduction.
This paper is concerned with KH[0, 1], the space of all functions f : [0, 1] [arrow right] R that are KH integrable (also known as Kurzweil, Henstock, Denjoy-Perron, gauge, nonabsolute, or generalized Riemann integrable). We emphasize that we are considering individual functions, whereas most of the related literature deals with KH[0, 1], the space of equivalence classes of KH integrable functions. Equivalence in this context means agreement outside some set of Lebesgue measure zero.
Sections 2 and 3 review basic results about the KH integral, including definitions of technical terms (gauges, «, f (T), etc.) used in this introduction. The KH integral generalizes the Lebesgue integral - we have the spaces of
functions KH[0, 1] ⊇ L^sup 1^[0, 1], and
equivalence classes of functions KH[0, 1] ⊇ L^sup 1^[0, 1].
In fact, both of those inclusions are strict; KH[0, 1] \ L^sup 1^[0, 1] contains erratic functions such as t^sup -1^ sin(t^sup -2^). But consequently the spaces KH and KH are also somewhat erratic. Apparently they cannot be equipped with topologies as nice as that of the Banach space L^sup 1^[0, 1].
Our notion of "niceness" is subjective, but can be formulated imprecisely as follows. A "nice" topology on a function space should be fairly simple to describe, should enjoy as many positive functional analytic properties (normability, completeness, etc.) as possible, and should be closely related to the properties (such as convergence) being studied for the functions involved. Without that requirement about convergences or other properties, we could...