The Birkhoff integral and the property of Bourgain

Scalar Boundedness of Vector-Valued Functions

“…This notion of integrability lies strictly between Bochner and Pettis integrability and has interesting features, see for example [2,9]. For instance, it can be characterised via the Bourgain property of certain families Z f,A where A ⊂ X * , see [2,15].…”

“…For instance, it can be characterised via the Bourgain property of certain families Z f,A where A ⊂ X * , see [2,15]. Following [14], we say that a family H ⊂ ‫ޒ‬ has the Bourgain property if for every ε > 0 and every E ∈ with μ(E) > 0 there are E 1 , .…”

On the Birkhoff Integral of Fuzzy Mappings in Banach Spaces

“…Birkhoff integrability lies strictly between Bochner and Pettis integrability when the range space X is nonseparable [2,8]. Lately, Several authors [4,7,9] have investigated the Birkhoff integral for Banach space valued functions. Several types of integrals of set-valued mappings were introduced by many authors.…”

Convergence theorems for the Birkhoff integral

“…The Birkhoff integral for Banach space valued functions, located strictly between the Bochner and Pettis integrals, was introduced in 1935 (see [1]). Lately, it has been investigated by several authors [2], [12], [9], [4], [10], [11]. A generalized version of the Birkhoff integral, invented by Dobrakov, has been studied in another recent article [13].…”

“…These equivalences were proved by B. Cascales and J. Rodríguez [2] (they assumed µ(Ω) = 1 but the theorem works for a σ-finite measure) and, independently, by the second author [10]. If Π and Γ are partitions of Ω, we say that Γ is finer than Π if each set from Γ is contained in some set from Π.…”