Convergence theorems for the Birkhoff integral

Pointwise limits of Birkhoff integrable functions

“…Following [1], we say that a collection {f n : n ∈ N} of Birkhoff integrable functions from Ω to X is equi-Birkhoff integrable if for every ε > 0 there is a countable partition (A m ) of Ω in Σ such that, for any choice of points t m ∈ A m , we have:…”

“…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.…”

“…In [16] a similar example is constructed, where the Banach space in the range is now any super-reflexive space with density character greater than or equal to c. As regards "positive" results, we have shown in [16] that Vitali's convergence theorem holds whenever the Banach space in the range is isomorphic to a subspace of ∞ . On the other hand, M. Balcerzak and M. Potyra la [1] have provided conditions ensuring the interchange of the operations of limit and Birkhoff integral which involve the notions of equi-Birkhoff integrability and almost uniform convergence.…”

On the Birkhoff Integral of Fuzzy Mappings in Banach Spaces

“…is made up of absolutely convergent series, then f : Ω → X is said to be absolutely Birkhoff integrable on Ω [1].…”

“…From [1,Theorem 7] and [1, Corollary 8], we can obtain the following two theorems using the same method in the Theorem 3.9.…”