Bartle–Dunford–Schwartz integration

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 L 0 ( μ ) the space of μ -measurable R -valued functions. Then all bounded measurable functions are μ -integrable (in an elementary sense), and the space L 1 ( 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 L 1 ( μ ) that is solid in L 0 ( μ ) . We prove general convergence theorems for both L 1 ( 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 c 0 , 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.