In 1931 H.F. Bohnenblust and E. Hille published a very important paper in which not only did they solve a long standing problem on convergence of Dirichlet series, but also gave a~general version of a celebrated inequality of Littlewood. Although it is full of extremely valuable mathematical ideas, the paper has been overlooked for a~long time and even today we feel that it does not get the credit it deserves. This may be caused by the not always accessible style that makes that the ideas are sometimes hidden. It is our intention to try to study the paper from a~modern point of view and to bring to light the valuable aspects we believe it has.

We deal with decomposition theorems for measures on Boolean algebras or -- more generally -- for modular functions on lattices. In particular, for measures on Boolean algebras with values in locally solid $\ell$-groups we compare decomposition theorems obtained with the Frechét-Nikodým-approach and decomposition theorems obtained using the band decomposition theorem of Riesz.

Spaces of scalar functions that are integrable in the sense of Bartle-Dunford-Schwartz integration, with respect to a~convexly bounded vector measure $\mu$, are studied. For instance, under the assumption that the range space $X$ of $\mu$ is sequentially complete, the effect of the Orlicz-Pettis property (with respect to a weaker topology on $X$) on the size of $L^1(\mu)$ is investigated. Some completeness properties of the space $L^1_\bullet(\mu)$ of `scalarly integrable functions' are established for general $X$.

We propose an essential improvement of a convergence theorem for the Birkhoff integral. We also obtain the respective version of this result for the convergence associated with an ideal on $\mathbb N$.

We present a complete characterization of finitely additive interval measures with values in conjugate Banach spaces which can be represented as Henstock-Kurzweil-Gelfand integrals. If the range space has the weak Radon-Nikodým property (WRNP), then we precisely describe when these integrals are in fact Henstock-Kurzweil-Pettis integrals.

We are concerned with the notions of absolute continuity in terms of $0$-continuity, and of $(\epsilon,\delta)$-continuity, with respect to a finitely additive function taking values into a topological commutative group. A sufficient condition for their equivalence is established. The non-additive case of functions with values into Hausdorff topological spaces is also investigated.

Let $\Sigma$ be a $\sigma$-algebra of subsets of a non-empty set $\Omega$. Let $B(\Sigma)$ be the space of all bounded $\Sigma$-measurable scalar functions defined on $\Omega$, equipped with the natural Mackey topology $\tau(B(\Sigma),ca(\Sigma))$. Let $(E,\xi)$ be a quasicomplete locally convex Hausdorff space and let $ca(\Sigma,E)$ be the space of all $\xi$-countably additive $E$-valued measures on $\Sigma$, provided with the topology ${\cal T}_s$ of simple convergence. We characterize relative ${\cal T}_s$-compactness in $ca(\Sigma,E)$, in terms of the topological properties of the corresponding sets in the space ${\cal L}_{\tau,\xi}(B(\Sigma),E)$ of all $(\tau(B(\Sigma),ca(\Sigma)),\xi)$-continuous integration operators from $B(\Sigma)$ to $E$. A generalized Nikodym type convergence theorem is derived.

Let $X$ be a real normed vector space and $\mathcal{B}$$(X)$ be the cone of all nonempty bounded closed convex subsets of $X$. For $A,B,C,D\in \mathcal{B}$$(X)$ we have a relation of equivalence defined by $(A,B)\sim (C,D)$ if and only if $\overline{A+D}=\overline{B+C}$. By $[A,B]$ we denote the quotient class of $(A,B)$. The quotient space $\widetilde{X}=\mathcal{B}$$^{2}(X)/_{\sim} $ is a vector space called the Minkowski-Rådström-Hörmander space over $X$. For $\tilde{x}=[A,B]\in \widetilde{X}$ we have the Hausdorff norm $\|\tilde{x}\|_{H}=d_{H}(A,B)=\inf \{\varepsilon >0|\, A\subset B+\varepsilon \mathbb{B},B\subset A+\varepsilon \mathbb{B}\}$ where $\mathbb{B}$ is the closed unit ball in $X$. We also define Bartels-Pallaschke norm $\|\tilde{x}\|_{BP}=\inf \{\|C\|+\|D\|\,|\,(C,D)\in [A,B] \}$, where $\|A\|=\sup_{a\in A}\|a\|$. In this paper we prove that the bilinear function $(\cdot,\cdot):(\widetilde{\mathbb{R}^{2}},\|\cdot\|_{H})\times (\widetilde{\mathbb{R}^{2}},\|\cdot\|_{BP}) \longrightarrow \mathbb{R}$ defined by $(\tilde{x},\tilde{y})=2V(\tilde{x},\tilde{y})+\langle s\tilde{x},s\tilde{y}\rangle$, where $V(\tilde{x},\tilde{y})$ is a generalized mixed volume and $s\tilde{x}$ is a generalized Steiner's point, satisfies the inequality $|(\tilde{x},\tilde{y})|\leq (2\pi +1)\|\tilde{x}\|_{H}\|\tilde{y}\|_{BP}$. We also prove that this bilinear function defines an isomorphic mapping between Banach spaces $(\widetilde{\mathbb{R}^{2}},\|\cdot \|_{BP})$ and the dual space to $(\widetilde{\mathbb{R}^{2}},\|\cdot \|_H)$ (Theorem 2).