Bartle-Dunford-Schwartz integration

We study the BDS-integral, using the original definition, but with respect to a convexly bounded measure mu with values in an arbitrary sequentially complete tvs X. Denote by L-0(mu) the space of mu-measurable R-valued functions. Then all bounded measurable functions are mu-integrable (in an elementary sense), and the space L-1(mu) of BDS-integrable functions is a vector lattice and a topological vector space in its natural topology. We next distinguish the space L-o(1)(mu) as the largest vector subspace of L-1(mu) that is solid in L-0(mu). We prove general convergence theorems for both L-1(mu) and L-o(1)(mu). In particular, we show that (4) with its natural topology is a Dedekind sigma-complete topological vector lattice with the sigma-Lebesgue property, and that the Dominated Convergence Theorem holds in L-o(1)(mu). If X contains no isomorphic copy of c(0), then L-o(1)(mu) has the sigma-Levi property (that is, the Beppo Levi Theorem holds). We identify L-o(1)(mu) as the domain for the Thomas-Turpin integral, and thus show that this integral is simply the restriction of the BDS-integral to L-o(1)(mu). The last two statements answer the questions posed, respectively, by Thomas and Turpin, and left open since the seventies. (C) 2012 Elsevier Inc. All rights reserved.