Uniform Convergence Theorem for the H1-Integral Revisited

Abstract

In this note we show that the uniform convergence theorem for the H 1 -integral is false.

Adjoint classes of functions in the H1 sense

“…After the original paper [6], a few further publications appeared [4], [5], [7], [8], [12], [13], and the H 1 -integral is already quite thoroughly investigated. Since the H 1 -integral is in fact a gauge integral (with the only difference in defining the limit of integral sums in slightly stronger terms), its theory helps to understand better the influence of gauges on Riemann-type integration.…”

“…However, it is the only convergence theorem which is known for this integral. The Beppo Levi (montone convergence) [12], the Lebesgue (dominated convergence) [12], and even the uniform convergence [7] theorems do not hold. Not every derivative is H 1 -integrable, but every derivative is the limit of a uniformly convergent sequence of H 1 -integrable functions [8].…”