On the limits of generalization of the Kolmogorov integral

Relations among Gauge and Pettis integrals for cwk(X)-valued multifunctions

“…As pointed out before, in case of single valued functions, according to [41] and [39, Remark 1], M-integrability is equivalent to the Birkhoff integrability. Definition 2.5.…”

“…In the family of the gauge integrals there is also the McShane integral and the versions of the Henstock and the McShane integrals when only measurable gauges are allowed (H and M integrals, respectively), and the variational Henstock and the variational McShane integrals. Moreover according to [41] and [39,Remark 1], the Birkhoff integral is a gauge integral too and it turns out to be equivalent to the M integral.…”

On the equivalence of McShane and Pettis integrability in non-separable Banach spaces

“…In general, the McShane integral lies strictly between the Bochner and Pettis integrals [13,17], although for some classes of Banach spaces McShane and Pettis integrability coincide: this happens for separable spaces [13,17,18], super-reflexive (e.g., Hilbert) spaces [4] and c 0 (Γ ) (where Γ is any non-empty set) [4]. The relationship between the McShane integral and others which are less known (e.g., the Henstock-Kurzweil, Birkhoff and Talagrand integrals) has been discussed in [11,12,27] and [29].…”

Pointwise limits of Birkhoff integrable functions

“…The Birkhoff integral [3] for functions taking values in Banach spaces plays an important role within the modern theory of vector integration, as has been noted recently in [1], [4], [5], [7], [16], [17], [18], [19], [20], [21] and [22] among others. An intriguing point concerns the validity of the classical convergence theorems of Lebesgue's integration theory in the case of Birkhoff integrable functions.…”