The Lebesgue-Nikodým Theorem states that for a Lebesgue measure λ:Σ〖⊂2〗^Ω→[0,∞) an additive set function F:Σ→R which is λ-absolutely continuous is the integral of a Lebegsue integrable a measurable function ...

We introduce the notion of summable bases that naturally generalizes the notion of unconditional sequence bases for Banach spaces. We shall be particularly interested in some classical results on sequences and series in ...

In this note, we propose a more efficient approach to the definition of the integral of Banach-valued functions that enables us to give an improved version of the Orlicz-Pettis theorem for integrals.

We use an extended theory of integral that generalizes the integration of vector valued functions with respect to non-negative, monotonic,countably subadditive set functions, in order to introduce a new approach to stochastic ...

We introduce a natural and more flexible approach to the definition of vector valued integral that will completely forgo any measurability assumption, strengthen the existing various classical concepts of ...

We give a non-metric version of the Besicovitch Covering Lemma and an extension of the Lebesgue-Differentiation Theorem into the setting of the integration theory of vector valued functions.

We revisit the definition of the tensor integrability introduced in [1], and prove some useful characterizations that provide a foundation for studying relevant properties to the integration theory. As applications, we ...

We study the equivalence property of the I- and II-L-Radon-Nikodým property (resp. I- and II-L-complete continuity property) types of Banach spaces. As a by-product, we obtain a proof of the following: F. Lust-Picard’s ...

A definitive positive answer to the so-called proper subspace problem (a.k.a. the atomic space problem) for quasi-normed spaces is given: every infinite dimensional quasi-normed space has a proper closed infinite-dimensional ...