ULTIMATELY INCREASING FUNCTIONS

作     者：Beer, Gerald Rodriguez-Lopez, Jesus 

作者机构：Calif State Univ Los Angeles Dept Math 5151 State Univ Dr Los Angeles CA 90032 USA Univ Politecn Valencia Inst Univ Matemat Pura & Aplicada E-46022 Valencia Spain 

出 版 物：《REAL ANALYSIS EXCHANGE》 

年 卷 期：2010年第36卷第1期

页      面：195-212页

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：Universidad Politecnica de Valencia (Programa de Apoyo a la investigacion y Desarrollo 2008) Generalitat Valenciana [GV/2007/198] 

主　　题：ultimately increasing function monotone convergence theorem increasing function subnet directed set chain order topology 

摘      要：A function g between directed sets = and = is called ultimately increasing if for each sigma(1) is an element of Sigma there exists sigma(2) = sigma(1) such that sigma = sigma(2) double right arrow g(sigma) = g(sigma(1)). A subnet of a net a defined on = [9] is nothing but a composition of the form a o g where g is ultimately increasing and g(Sigma) is a cofinal subset of Lambda. While even for linearly ordered sets, an increasing net defined on a cofinal subset of the domain need not have an increasing extension, in complete generality, it must have an ultimately increasing extension, and conversely when the domain is linearly ordered. Applications are given in the context of functions with values in a linearly ordered set equipped with the order topology - in particular, the extended real numbers. For example, we show that a real sequence converges to the supremum of its set of terms if and only if is the supremum of the ultimately increasing sequences that it majorizes.