Bartle–Dunford–Schwartz integration

Abstract

We study the BDS-integral, using the original definition, but with respect to a convexly bounded measure μ with values in an arbitrary sequentially complete tvs X. Denote by L0(μ) the space of μ-measurable R-valued functions. Then all bounded measurable functions are μ-integrable (in an elementary sense), and the space L1(μ) of BDS-integrable functions is a vector lattice and a topological vector space in its natural topology. We next distinguish the space L∘1(μ) as the largest vector subspace of L1(μ) that is solid in L0(μ). We prove general convergence theorems for both L1(μ) and L∘1(μ). In particular, we show that L∘1(μ) with its natural topology is a Dedekind σ-complete topological vector lattice with the σ-Lebesgue property, and that the Dominated Convergence Theorem holds in L∘1(μ). If X contains no isomorphic copy of c0, then L∘1(μ) has the σ-Levi property (that is, the Beppo Levi Theorem holds). We identify L∘1(μ) as the domain for the Thomas–Turpin integral, and thus show that this integral is simply the restriction of the BDS-integral to L∘1(μ). The last two statements answer the questions posed, respectively, by Thomas and Turpin, and left open since the seventies.