In this paper we present some new versions of Brooks-Jewett and Dieudonné-type theorems for $\left(l\right)$-group-valued measures.

Nell'ultimo ventennio tutta una serie di lavori è stata rivolta allo studio delle misure su strutture algebriche più generali delle algebre di Boole, come i poset e i reticoli ortomodulari, le effect algebras, le BCK-algebras. La teoria così ottenuta interessa l'analisi funzionale, il calcolo delle probabilità e la topologia, più recentemente la teoria delle decisioni. Si presentano alcuni risultati relativi a misure su strutture algebriche non-standard analizzando, in particolare, gli aspetti topologici...

After a short discussion of the first application of measure theoretic tools to economics we show that it is consistent relative to the usual axioms of set theory that there exists no nonatomic probability space of power less than the continuum. This together with other results shows that Aumann's continuum-of-agents methodology provides a sound framework at least for the cooperative theory. There are, however, other problems in economics where, without further assumptions, the continuum may be...

Let $M(X,𝒜)$ ($M^{*}(X,𝒜)$) be the $f$-ring of all (bounded) real-measurable functions on a $T$-measurable space $(X,𝒜)$, let $M_K(X,𝒜)$ be the family of all $f\in M(X,𝒜)$ such that $\mathrm{coz}\left(f\right)$ is compact, and let $M_{\infty }(X,𝒜)$ be all $f\in M(X,𝒜)$ that $\{x\in X:|f\left(x\right)|\ge \frac{1}{n}\}$ is compact for any $n\in \mathbb{N}$. We introduce realcompact subrings of $M(X,𝒜)$, we show that $M^{*}(X,𝒜)$ is a realcompact subring of $M(X,𝒜)$, and also $M(X,𝒜)$ is a realcompact if and only if $(X,𝒜)$ is a compact measurable space. For every nonzero real Riesz map $\varphi :M(X,𝒜)\to ℝ$, we prove that there is an element $x_0\in X$ such that $\varphi \left(f\right)=f\left(x_0\right)$ for every $f\in M(X,𝒜)$ if $(X,𝒜)$ is a compact measurable space. We confirm...

Let g be a doubling gauge. We consider the packing measure ${}^g$ and the packing premeasure ${₀}^g$ in a metric space X. We first show that if ${₀}^g\left(X\right)$ is finite, then as a function of X, ${₀}^g$ has a kind of “outer regularity”. Then we prove that if X is complete separable, then $\lambda sup{₀}^g\left(F\right){\le }^g\left(B\right)\le sup{₀}^g\left(F\right)$ for every Borel subset B of X, where the supremum is taken over all compact subsets of B having finite ${₀}^g$-premeasure, and λ is a positive number depending only on the doubling gauge g. As an application, we show that for every doubling gauge...

We present the full descriptive characterizations of the strong McShane integral (or the variational McShane integral) of a Banach space valued function $f:W\to X$ defined on a non-degenerate closed subinterval $W$ of ${ℝ}^m$ in terms of strong absolute continuity or, equivalently, in terms of McShane variational measure $V_{ℳ}F$ generated by the primitive $F:{ℐ}_W\to X$ of $f$, where ${ℐ}_W$ is the family of all closed non-degenerate subintervals of $W$.