Abstract integration with respect to measures and applications to modular convergence in vector lattice setting

Abstract

convergences, satisfying suitable axioms, and some fundamental properties are studied. Moreover, by means of this integral, some convergence results on operators in vector lattice-valued modulars are proved. Some applications are given to moment kernels and to the Brownian motion.

Some applications of modular convergence in vector lattice setting

“…The results obtained are extensions of those on the integrals investigated in [14], where only finite measures are considered, and in [16], where the vector lattices considered there have suitable properties. In this paper we continue the investigation started in [19], and give examples and applications on moment and Mellin-type kernels, as well as linear and nonlinear vector lattice-valued operators.…”

“…For what unexplained we refer, for instance, to [14,19,[33][34][35][36][37]. Now we recall the axioms given in [19] in order to present the abstract convergence in the vector lattice context (see, e.g., [14,Definition 2.1] and [3]).…”

“…We denote by S the set of all simple functions defined on G, which assume a finite number of values and with support contained in a set of finite measure μ. For such functions, the integral is defined in a natural way (see, e.g., [14,19]). Now, to introduce the integral for more general functions, we use the definitions of uniform convergence and convergence in measure and in L 1 .…”

“…In this paper we continue the study started in [3,10,11,[16][17][18][19] were we have extended to the vector lattice setting the problem of approximating a function f by means of Urysohn-type integral operators in the setting of modular convergence. These operators are particularly useful in order to approximate a continuous or analog signal by means of discrete samples, and therefore they are widely applied for instance in reconstructing images, see for example [1,2,5,6,[29][30][31][32].…”

“…We met him in 1990, in Capri during the conference "Quarto Convegno di Analisi Reale e Teoria della Misura", in which he presented the results of [23] in the main conference "The sampling theorem and its unique role in various branches of mathematics", which has influenced our research in this area and related topics and applications. In [19] we have considered vector lattice-valued functions defined on a metric space, vector lattice-valued measures and we have constructed a vector lattice-valued integral, involving a triple of vector lattices, linked by a suitable "product" structure. The results obtained are extensions of those on the integrals investigated in [14], where only finite measures are considered, and in [16], where the vector lattices considered there have suitable properties.…”