In this paper we propose some propositions in reference to the notion of exhaustiveness which applies for families and sequences of functions. This new notion given for the first time from Gregoriades and Papanastassiou has a good progress and in mathematical community it is followed by many papers. We are concerned in study these families in framework of exhaustiveness in order to apply to measure theory and Henstock-Kurzweil integration. We investigate the properties of one well known convergence, local uniformly convergence or short "-convergence" and compare it with other convergences.
This paper studies the construction of three functional topologies in 𝑌^𝑋 production spaces. Their topological bases have been found and compared between them. An important place is a comparison with wellknown topologies such as uniform, point, and open compact convergence. The convergences used are uniform local convergence, strongly uniformed local convergence, and the convergence known as α-convergence.
The main goal of this article is to investigate the integration theory of Mcshane type for functions with values in ordered spaces. Afected by the work of Boccuto, Riecan, and Vrábelová with Kurzweil-Henstock integration we studied the same problems for another important type of integration on such space as Mcshane ones.In this paper we present in other way the definition of Mcshane integral on the Riesz space using a very important lemma of famous Fremlin. In the second section we reconstruct allmost all the propeties of Mchane integral given in [5], [6] and these ones become a little more stronger. We arrive some new results compared with Henstock-Kurzweil ones. In the third section we define the strong version of Mcshane integral and give the neccesary and sufficent condition of this concept. In the fourth section we prove the fundamental theorems of Calculus for the D-ܯintegral and in the fifth we extended an application of this integration to Walsh series .Mathematics Subject Classification: 28B15, 28B05, 28A39, 42C10, 42C25, 46G10
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