Let $X$ be a quasicomplete locally convex Hausdorff space. Let $T$ be a locally compact Hausdorff space and let $C_0\left(T\right)=\{fT\to I$, $f$ is continuous and vanishes at infinity$\}$ be endowed with the supremum norm. Starting with the Borel extension theorem for $X$-valued $\sigma$-additive Baire measures on $T$, an alternative proof is given to obtain all the characterizations given in [13] for a continuous linear map $u{C}_0\left(T\right)\to X$ to be weakly compact.

We slightly modify the definition of the Kurzweil integral and prove that it still gives the same integral.

We show that a Pettis integrable function from a closed interval to a Banach space is Henstock-Kurzweil integrable. This result can be considered as a continuous version of the celebrated Orlicz-Pettis theorem concerning series in Banach spaces.

We study a generalization of the classical Henstock-Kurzweil integral, known as the strong $\rho$-integral, introduced by Jarník and Kurzweil. Let $({𝒮}_{\rho }\left(E\right),\parallel ·\parallel )$ be the space of all strongly $\rho$-integrable functions on a multidimensional compact interval $E$, equipped with the Alexiewicz norm $\parallel ·\parallel$. We show that each element in the dual space of $({𝒮}_{\rho }\left(E\right),\parallel ·\parallel )$ can be represented as a strong $\rho$-integral. Consequently, we prove that $fg$ is strongly $\rho$-integrable on $E$ for each strongly $\rho$-integrable function $f$ if and only if $g$ is almost everywhere...

For Banach-space-valued functions, the concepts of 𝒫-measurability, λ-measurability and m-measurability are defined, where 𝒫 is a δ-ring of subsets of a nonvoid set T, λ is a σ-subadditive submeasure on σ(𝒫) and m is an operator-valued measure on 𝒫. Various characterizations are given for 𝒫-measurable (resp. λ-measurable, m-measurable) vector functions on T. Using them and other auxiliary results proved here, the basic theorems of [6] are rigorously established.

In the present work we prove that, in the space of Pettis integrable functions, any subset that is decomposable and closed with respect to the topology induced by the so-called Alexiewicz norm $\left|∥·∥\right|$ (where $\left|∥f∥\right|={sup}_{[a,b]\subset [0,1]}\parallel {\int }_a^{b}f\left(s\right)ds\parallel$) is convex. As a consequence, any such family of Pettis integrable functions is also weakly closed.

If $(\Omega ,\Sigma ,\mu )$ is a finite measure space and $X$ a Banach space, in this note we show that $L_{w^{*}}^1(\mu ,X^{*})$, the Banach space of all classes of weak* equivalent $X^{*}$-valued weak* measurable functions $f$ defined on $\Omega$ such that $\parallel f\left(\omega \right)\parallel \le g\left(\omega \right)$ a.e. for some $g\in L_1\left(\mu \right)$ equipped with its usual norm, contains a copy of $c_0$ if and only if $X^{*}$ contains a copy of $c_0$.

If E is a Banach space with a basis {en}, n belonging to N, a vector measure m: a --> E determines a sequence {mn}, n belonging to N, of scalar measures on a named its components. We obtain necessary and sufficient conditions to ensure that when given a sequence of scalar measures it is possible to construct a vector valued measure whose components were those given. Furthermore we study some relations between the variation of the measure m and the variation of its components.