Estudiamos cuando el límite uniforme de una red de funciones cuasi-continuas con valores en un espacio localmente convexo X es también una función cuasi-continua, resaltando que esta propiedad depende del menor cardinal de un sistema fundamental de entornos de O en X, y estableciendo condiciones necesarias y suficientes. El principal resultado de este trabajo es el Teorema 15, en el que los resultados de [7] y [10] son mejorados, en relación al Teorema de L. Schwartz.

A Boolean algebra $𝒜$ has the interpolation property (property (I)) if given sequences $\left(a_n\right)$, $\left(b_m\right)$ in $𝒜$ with $a_n\le b_m$ for all $n,m$, there exists an element $b$ in $𝒜$ such that $a_n\le b\le b_n$ for all $n$. Let $𝒜$ denote an algebra with the property (I). It is shown that if $({\mu }_n:𝒜\to X)$ ($X$ a Banach space) is a sequence of strongly additive measures such that ${lim}_n{\mu }_n\left(a\right)$ exists for each $a\in 𝒜$, then $\mu \left(a\right)={lim}_n{\mu }_n\left(a\right)$ defines a strongly additive map from $𝒜$ to $X$and the ${\mu }_n^{\text{'}}s$ are uniformly strongly additive. The Vitali-Hahn-Saks (VHS) theorem for strongly additive $X$-valued measures defined...

In [4, 5, 7] an abstract, versatile approach was given to sequential weak compactness and lower closure results for scalarly integrable functions and multifunctions. Its main tool is an abstract version of the Komlós theorem, which applies to scalarly integrable functions. Here it is shown that this same approach also applies to Pettis integrable multifunctions, because the abstract Komlós theorem can easily be extended so as to apply to generalized Pettis integrable functions. Some results in the...

We show that if T is an uncountable Polish space, 𝓧 is a metrizable space and f:T→ 𝓧 is a weakly Baire measurable function, then we can find a meagre set M ⊆ T such that f[T∖M] is a separable space. We also give an example showing that "metrizable" cannot be replaced by "normal".

Let $ℬ=f:ℝ\to ℝ:fisbounded$, and $ℳ=f:ℝ\to ℝ:fisLebesguemeasurable$. We show that there is a linear operator $\Phi :ℬ\to ℳ$ such that Φ(f)=f a.e. for every $f\in ℬ\cap ℳ$, and Φ commutes with all translations. On the other hand, if $\Phi :ℬ\to ℳ$ is a linear operator such that Φ(f)=f for every $f\in ℬ\cap ℳ$, then the group $G_{\Phi }$ = a ∈ ℝ:Φ commutes with the translation by a is of measure zero and, assuming Martin’s axiom, is of cardinality less than continuum. Let Φ be a linear operator from ${ℂ}^{ℝ}$ into the space of complex-valued measurable functions. We show that if Φ(f) is non-zero for every $f\left(x\right)=e^{cx}$, then $G_{\Phi }$ must...

In this paper we bring together the different known ways of establishing the continuity of the integral over a uniformly integrable set of functions endowed with the topology of pointwise convergence. We use these techniques to study Pettis integrability, as well as compactness in C(K) spaces endowed with the topology of pointwise convergence on a dense subset D in K.

Pour tout compact complètement régulier $T$, on désigne par $M_{\beta }\left(T\right)$ l’espace des mesures de Radon sur le compactifié de Stone-Cech $\beta T$ de $T$ et par $M_{\sigma }\left(T\right)$ son sous-espace formé des mesures $\sigma$-régulières au sens de Varadarajan. On décrit alors sur ces deux espaces des topologies ${\mathbf{T}}_p$, $1\le p\le +\infty$, qui possèdent des propriétés curieuses parmi lesquelles il convient de citer la suivante : pour $1\<p\le +\infty$ et pour tout $T$ non pseudocompact, l’espace $\left(M_{\sigma }\right(T),{\mathbf{T}}_p)$ est non quasi-complet mais ses précompacts sont relativement compacts. Ce résultat permet...