Some Comments on the <em>H</em>₁-Integral

Integration with One-Dimensional Space of Gauges

“…In this connection, one can find the H 1 -integral interesting not as being an alternative tool for integration, with some advantages over other integration processes, but as producing the class of H 1 -integrable functions with its relations to other classes of functions. A few important results have been received in the latter direction, but, it seems, some investigation opportunities remain, especially those related to characterizations of classes generated by some arithmetic operations with H 1 -integrable functions as one of the arguments (for a sample problem see [12,Question 2.3]).…”

“…Denote T s = {(I, x) ∈ S s : I ⊂ O}. For each (I, x) ∈ T s we have I ∩ D = ∅, whence by the definition of O, |∆|G(T s ) < ε and so (x ∈ D) (12) |σ…”

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.…”

“…There are Lebesgue integrable functions not integrable in the H 1 sense [12], and there are H 1 -integrable functions not integrable in the sense of Lebesgue (this is so because the H 1 -integral is not absolute) [4], [12]. There is a Kurzweil-Henstock integrable function equal almost everywhere to no H 1 -integrable one [8], but every Kurzweil-Henstock integrable function can be written as the sum of a Lebesgue integrable one and an H 1 -integrable one [13]. A controlled convergence theorem for the H 1 -integral was proved in [5].…”