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

Abstract

We show that McShane and Pettis integrability coincide for functionswhere μ is any finite measure. On the other hand, assuming the Continuum Hypothesis, we prove that there exist a weakly Lindelöf determined Banach space X, a scalarly null (hence Pettis integrable) function h : [0, 1] → X and an absolutely summing operator u from X to another Banach space Y such that the composition u • h : [0, 1] → Y is not Bochner integrable; in particular, h is not McShane integrable.

Integration in Hilbert generated Banach spaces

“…Cases (ii) and (iii) are due to Di Piazza and Preiss [6]. Case (iv) was recently proved by the second-named author [27]. (i) X is separable.…”

“…In this way, they asked [6, p. 1178] whether McShane and Pettis integrability are equivalent for functions taking values in arbitrary WCG Banach spaces. The second-named author [27] gave another partial answer to this question by showing that this is always the case for L 1 (ν) (where ν is any probability measure).…”

“…Under the Continuum Hypothesis, Di Piazza and Preiss [6] and the second-named author [27] provided examples of scalarly null functions defined on [0, 1] which are not McShane integrable.…”

Multifunctions determined by integrable functions

“…Instead, a generalization of the Lebesgue integral's definition by using Riemann sums, produces the McShane and the Birkhoff integral for vector valued functions. If the Banach space is separable, then the Pettis, McShane and Birkhoff integrals concide; but for more general Banach spaces they are in general different (see also [25,42]). In vector spaces it is also possible to give a version of Choquet integral, also combining it with Pettis integral (cf.…”

On p-Dunford integrable functions with values in Banach spaces

“…As remarked in [7, p. 56], his argument can be easily modified for arbitrary 1 ≤ p < ∞ to obtain that u • f is p-Bochner integrable whenever f is strongly measurable and p-Dunford integrable. Several papers discussed such type of questions for p = 1 beyond the strongly measurable case, see [3,4,19,31,32]. An unpublished result of Lewis [19], rediscovered independently in [31,Theorem 2.3], states that for p = 1 the composition u • f is at least scalarly equivalent to a Bochner integrable function if f is Dunford integrable.…”