MR3266136 Porcello, G., Decomposability in the space of HKP-integrable functions. Math. Nachr. 287 (2014), no. 14-15, 17331744. 26A39 (26E25 28B20 54C60)

The notion of decomposability for families of Banach space valued functionsis a certain kind of generalization of convexity. Decomposability is usually de- ned (in a space, or some subspaces, of measurable functions as the space ofBochner integrable or Pettis integrable functions) with respect to a -algebraof sets. In the paper under review the author introduces the notion of decom-posability for vector-valued functions integrable in Henstock sense. Since theHenstock-type integrals act only on intervals, the author modi es in a slightbut essential way the classical"de nition of decomposability: instead of a -algebra of sets, one has to work with the ring A generated by the subintervals[a; b) [0; 1].Let X be a separable Banach space and let HKP([0; 1];X) be the familyof all Henstock-Kurzweil-Pettis integrable (for short HKP-integrable) functionsfrom [0; 1] to X. A set K HKP([0; 1];X) is said to be decomposable withrespect to the ring A if for all f1; f2 2 K and for all E 2 A, f1 E +f2 Ec 2 K.The main results of the paper are two representation theorems for decom-posable sets of HKP-integrable or Henstock integrable functions (see Theorem4.4 and Theorem 6.1, respectively). The author proves that if X has the Schurproperty (or does not contain any isomorphic copy of c0), then each decompos-able, convex and closed in the Alexiewicz norm subset of HKP-integrable (orHenstock integrable) functions that satis es some mild additional conditions,can be represented as the set of the HKP-integrable (or Henstock integrable)selections of a suitable multifunction. This is a generalization of earlier results ofGodet-Thobie, and Satco [Quaest. Math. 29 (2006), no. 1, 39-58, MR2209790]and of Chakraborty and Choudrhury [J. Convex Anal. 19 (2012), no. 3, 671-683,MR3013754] for decomposable sets in the family of X valued Pettis integrablefunctions.