arXiv subjects

Alexandre Reggiolli Teixeira

Publications and source records attributed to Alexandre Reggiolli Teixeira.

5 recordsLinked to original sources

arXiv2026

Measure Theory in Convergence Spaces Part II: Abstract Integration in Net Convergence Structures

The construction of a coherent and general integration procedure is the central question of this second part of our study of measure theory in convergence spaces. We define, and establish convergence theorems for such process of integration, called the net Riemann integration, in convergence groups and commutative monoids, preserving the prescribed convergence throughout. The underlying interchange theorem admits noncommutative groups and requires right completeness only at the cardinality of the error family. Under intrinsic stability assumptions, countable error control therefore needs only sequential completeness, even for arbitrary integration nets, from which we also stablish a general version of Henstock's lemma. The resulting uniform theorems reaches incomplete ranges in which no complete subspace contains a tail of the approximating integrals. In the ordered locally solid convergence theory side, countability of the inner direction replaces countable accessibility of the approximants, and local existence arguments yield monotone and dominated convergence and, under additional controller assumptions, a further extension through increasing integrable barriers with pointwise coverage and uniformly vanishing excess primitives. This constructs the limiting integral without a common integrable envelope and gives uniform control of primitive variation. We finish with two appendices on new and general results of abstract metrization of net convergences and related topics.

math.FA↗

arXiv2026

Measure Theory in Convergence Spaces Part I: Sion-Carathéodory Extension in Net Convergence Groups

We establish a Carathéodory-type extension theorem for premeasures with values in general separated abelian net convergence groups. Starting from hereditary exhaustivity on the source ring and intrinsic conditions for combining null errors, we construct a countably additive extension on a regulated Boolean sigma-algebra, with full upper and lower regularity in the original convergence. Completeness is required only on the limit-stable hull of the source range. Under closed summation for arbitrary directed rows and finite-response reflection, we identify the regulated domain with the splitting class of Sion's inner and outer processes. For a Dedekind-complete real vector lattice, we show that sequential closed summation on a bounded interval is equivalent to weak sigma-distributivity of the generated band, giving a converge-theoretic interpretation of the concept. Source representations allow us to recover the partially ordered-group extension of Volauf and the finite-variation extension of Kusraev and Malyugin from the general theorem. We then treat finite semivariation by adjoining dominated coordinate variations to an additive lift of the premeasure. The resulting extension is defined on the full semivariation regulator domain and preserves semivariation exactly, without requiring decomposability or the countable sup property. It includes measures of infinite vector variation.

math.FA↗

arXiv2026

Sewing on Thin Groupoids, Knitting, and Based Holonomy for Lipschitz Paths

An approximate action of the pair groupoid of a metric space determines an action of its Lipschitz-thin groupoid on complete extended metric fibers, uniquely characterized by its local sewing estimate. We prove this assertion under a superlinear three-point defect estimate and a Lipschitz bound for finite products. As a result, we obtain a positive proof of Curry and Manchon's sewing conjecture [2], with the necessary path-scaling correction to its estimate. Further restriction to isotropy constructs the based holonomy anticipated in their Remark 4.15 as a representation of based Lipschitz loops modulo thin equivalence. Rectangular comparison estimates give full relative Lipschitz-homotopy descent, in particular under their strong knitting hypothesis and under any three point estimate of total order greater than two. We prove the corresponding flatness and basepoint-covariance statements. An area model shows that the order threshold is sharp. We also separate the exponential estimate from the general knitting assumptions and exhibit a compact disk metric for which strong knitting does not imply invariance under merely continuous homotopy. As an application, for controlled fields on a metric space, we prove that the classical rough integral descends to the thin groupoid and admits substitution with a common rough controller.

math.FA↗

arXiv2026

Control Measures for Bochner $L_{0}$-Valued Vector Measures

It is shown that for any finite positive measure $μ$ defined on a measure space $(S, Σ)$, and any Banach or Fréchet space $Z$, the control measure Theorem of Talagrand (T) is true for the case when the (stochastic) vector measure $\boldsymbol{m}:\mathcal{E} \to L_0(μ,Z)$, defined on another measurable space $(E, \mathcal{E})$, takes values in $L_{0}(μ,Z)$, the Bochner space of vector-valued functions associated to $μ$ and $Z$. As a consequence, we also obtain a Rybakov type result for this control. Finally, we give the relation of this result to bounded multiplier properties (BMP) of $F$-spaces and pose various open problems related to it.

math.FA↗

arXiv2026

Abstract Integration in Net Convergence Structures

In this article, we propose a general theory of integration of the Riemann and Lebesgue types with respect to arbitrary measures and functions, connected by a continuous bilinear product, with values in abstract vector spaces endowed with a convergence structure given by nets. This covers both the topological and order based convergences in the literature. We then show that this integral satisfies most of the common properties of the objects that comprises integration theory. By establishing a generalized notion of summability on Riesz spaces and an integral built upon countable partitions of the base space, we then stablish some uniform, monotone and dominated convergence theorems for the refereed integrals, as well as a non-topological or order based Henstock Lemma and a general convergence theorem based on the notion of conjugated lattice seminorms. An application of these theorems is made to prove various equivalences concerning the Lebesgue, for which we give a brief survey, Saks and Riemann type integrals in partially ordered and topological vector spaces presented in the literature, for which we also make a thorough review. We finish the article with a possible way of classifying general integration procedures defined in abstract convergence structures, and pose some open problems based on them.

math.FA↗