The notion of a topological vector space of Bochner measurable functions is introduced and studied. Among the main results obtained are characterizations of completeness and of containment of copies of c(0) or l(infinity).

The area of research of this paper goes back to a 1930 result of H. Auerbach showing that a scalar series is (absolutely) convergent if all its sere-density subseries converge. A series Sigma(n)x(n) in a topological vector space X is called L-convergent if each of its lacunary subseries Sigma(k)x(nk) (i.e. those with n(k+1) - n(k) --> infinity) converges. The space X is said to have the Lacunary Convergence Property, or LCP, if every L-convergent series in X is convergent; in fact, it is then subseries convergent. The Zero-Density Convergence Property, or ZCP, is defined similarly though of lesser importance here. It is shown that for every L-convergent series the set of all its finite sums is metrically bounded; however, it need not be topologically bounded. Next, a space with the LCP contains no copy of the space c(0). The converse holds for Banach spaces and, more generally, sequentially complete locally pseudoconvex spaces. However, an F-lattice of measurable functions is constructed that has both the Lebesgue and Levi properties, and thus Contains no copy of c(0), and, nonetheless, lacks the LCP, The main land most difficult) result of the paper is that if a Banach space E contains no copy of c(0) and lambda is a finite measure, then the Bochner space L-0(lambda, E) has the LCP. From this, with the help of some Orlicz-Pettis type theorems proved earlier by the authors, the LCP is deduced for a vast class of spaces of (scalar and vector) measurable functions that have the Lebesgue type property and are "metrically-boundedly sequentially closed" in the containing L-0 space. Analogous results about the convergence of L-convergent positive series in topological Riesz spaces are also obtained. Finally, while the LCP implies the ZCP trivially, an example is given that the converse is false, in general.

For a finite measure lambda, let L-0(lambda) denote the space of lambda-measurable functions equipped with the topology of convergence in measure. We prove that a series in L-0(lambda) is subseries (or unconditionally) convergent provided each of its lacunary subseries converges.

A finitely additive vector measure from a σ \sigma -ring to a Riesz space is countably additive (exhaustive) for all Hausdorff Lebesgue topologies on the range space, or for none of them. In particular, subseries convergent series are the same for all Hausdorff Lebesgue topologies on a Riesz space.

The paper is concerned with order-topological characterizations of topological Riesz spaces, in particular spaces of measurable functions, not containing Riesz isomorphic or linearly homeomorphic copies of c 0 c_{0} or ℓ ∞ \ell _{\infty } .

It is shown that the proper domains of integral operators have separating duals but in general they are not locally convex. Banach function spaces which can occur as proper domains are characterized. Some known and some new results are given, illustrating the usefulness of the notion of proper domain.

AbstractLet (L, λ) be a locally solid Riesz space. (L, λ) is said to have the Levi property if for every increasing λ-bounded net (xα) ⊂ L+, sup xα exists. The Levi property, appearing in literature also as weak Fatou property (Luxemburg and Zaanen), condition (B) or monotone completeness (Russian terminology), is a classical object of investigation. In this paper we are interested in some variations of the property, their mutual relationships and applications in the theory of topological Riesz spaces. In the first part of the paper we clarify the status of two problems of Aliprantis and Burkinshaw. In the second part we study ideal-injective Riesz spaces.

The results of this paper clarify and extend slightly the previous work of Dolecki and Lechicki (C. R. Acad. Sci. Paris, 293 1981, 219–221; J. Math. Anal. Appl., 88 1982, 547–584) and Hansell, Jayne, Rogers and the author (Math. Z., 189 1985, 297–318). Let X, Y be Hausdorff spaces and F: X → Y an upper semicontinuous set-valued map. A subset K of F(x) is said to be a peak of F at x, if, for every open set V containing K, there exists a neighbourhood U of x such that F(U)/F(x)/t(V). Criteria (“Choquet-Dolecki Theorems”) are given in order that F has the smallest possible peak. It turns out that in unexpectedly general situations an upper semicontinuous map F has, for every x in X, a peak which is the smallest possible at x and moreover compact.

A necessary and sufficient condition for a Banach space to have on itself a nontrivial Saks space is given.