Several comments on the Henstock-Kurzweil and McShane integrals of vector-valued functions

On the integration of Riemann-measurable vector-valued functions

“…If f is H -integrable on [a, b] and its indefinite Henstock integral is AC on [a, b], then combining Theorem 4 of [28] and Theorem 3 of [26] shows that f must be both Riemann measurable and McShane integrable on [a, b]. Lastly, Theorem 7 of [28]…”

“…The next theorem provides a descriptive su cient condition for M -integrability on a closed set (cf. [26,Theorem 5]).…”

“…Our methods refine those in [26] so that we can get most of the results in a form that is as close as possible to the above real-valued case. Theorem 7 clarifies the relationship between two di↵erent absolute continuity concepts, the AC ⇤ and AC ⇤ properties, defined in terms of constant and measurable gauges, respectively.…”

“…The descriptive relationship between the Henstock and McShane integrals for vector-valued functions has received some attention in [7,26,27]. In [7], Fremlin proved that a function is McShane integrable if and only if it is both Henstock and Pettis integrable or equivalently, it is Henstock integrable and its indefinite Henstock integral is AC (see [26]).…”

“…The descriptive relationship between the Henstock and McShane integrals for vector-valued functions has received some attention in [7,26,27]. In [7], Fremlin proved that a function is McShane integrable if and only if it is both Henstock and Pettis integrable or equivalently, it is Henstock integrable and its indefinite Henstock integral is AC (see [26]). In [26], by means of Fremlin's criterion it was shown that in some classes of Banach spaces the domain of a Henstock integrable function can be written as a countable union of closed sets on each of which the integrand is McShane integrable and the corresponding indefinite McShane integrals converge to the indefinite Henstock integral in the Alexiewicz norm.…”