In this paper we define the derivative and the Denjoy integral of mappings from a vector lattice to a complete vector lattice and show the fundamental theorem of calculus.

Let ν be a positive measure on a σ-algebra Σ of subsets of some set and let X be a Banach space. Denote by ca(Σ,X) the Banach space of X-valued measures on Σ, equipped with the uniform norm, and by ca(Σ,ν,X) its closed subspace consisting of those measures which vanish at every ν-null set. We are concerned with the subsets ${}_{\nu }\left(X\right)$ and ${}_{\nu }\left(X\right)$ of ca(Σ,X) defined by the conditions |φ| = ν and |φ| ≥ ν, respectively, where |φ| stands for the variation of φ ∈ ca(Σ,X). We establish necessary and sufficient conditions...

We prove that the range of a vector measure determines the σ-finiteness of its variation and the derivability of the measure. Let F and G be two countably additive measures with values in a Banach space such that the closed convex hull of the range of F is a translate of the closed convex hull of the range of G; then F has a σ-finite variation if and only if G does, and F has a Bochner derivative with respect to its variation if and only if G does. This complements a result of [Ro] where we proved...

Some versions of Dieudonné-type convergence and uniform boundedness theorems are proved, for $k$-triangular and regular lattice group-valued set functions. We use sliding hump techniques and direct methods. We extend earlier results, proved in the real case. Furthermore, we pose some open problems.

Let $X$ be a completely regular Hausdorff space, $E$ a real normed space, and let $C_b(X,E)$ be the space of all bounded continuous $E$-valued functions on $X$. We develop the general duality theory of the space $C_b(X,E)$ endowed with locally solid topologies; in particular with the strict topologies ${\beta }_z(X,E)$ for $z=\sigma ,\tau ,t$. As an application, we consider criteria for relative weak-star compactness in the spaces of vector measures $M_z(X,E^{\text{'}})$ for $z=\sigma ,\tau ,t$. It is shown that if a subset $H$ of $M_z(X,E^{\text{'}})$ is relatively $\sigma (M_z(X,E^{\text{'}}),C_b(X,E))$-compact, then the set $conv\left(S\right(H\left)\right)$ is still relatively $\sigma (M_z(X,E^{\text{'}}),C_b(X,E))$-compact...

If $(\Omega ,\Sigma )$ is a measurable space and $X$ a Banach space, we provide sufficient conditions on $\Sigma$ and $X$ in order to guarantee that $\mathrm{b}vca(\Sigma ,X)$, the Banach space of all $X$-valued countably additive measures of bounded variation equipped with the variation norm, contains a copy of $c_0$ if and only if $X$ does.

We present two types of representation theorems: one for linear continuous operators on space of Banach valued regulated functions of several real variables and the other for bilinear continuous operators on cartesian products of spaces of regulated functions of a real variable taking values on Banach spaces. We use generalizations of the notions of functions of bounded variation in the sense of Vitali and Fréchet and the Riemann-Stieltjes-Dushnik or interior integral. A few applications using geometry...

Let X be a Banach space and ν a countably additive X-valued measure defined on a σ-algebra. We discuss some generation properties of the Banach space L¹(ν) and its connection with uniform Eberlein compacta. In this way, we provide a new proof that L¹(ν) is weakly compactly generated and embeds isomorphically into a Hilbert generated Banach space. The Davis-Figiel-Johnson-Pełczyński factorization of the integration operator $I_{\nu }:L¹\left(\nu \right)\to X$ is also analyzed. As a result, we prove that if $I_{\nu }$ is both completely continuous...