Topologizing the Denjoy space by measuring equiintegrability.

Abstract

Basic limit theorems for the KH integral involve equiintegrable sets. We construct a family of Banach spaces $X_\Delta$ whose bounded sets are precisely the subsets of ${\cal KH}[0,1]$ that are equiintegrable and pointwise bounded. The resulting inductive limit topology on $\bigcup_\Delta X_\Delta = {\cal KH}[0,1]$ 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.