On C₀-saturated Property Of LP Spaces And Borel Measures On L₁

A Banach space X is called a Lindenstrauss-Phelps space(LP space)if the set of extreme points extBX*of the unit ball BX*of X*is countable,and we denote ext BX*=(?).In 1978,V.P.Fonf introduced a new norm on X:Taking an arbitrary sequence ?n>0,?1<1.?n?0,define(?){(1+?n)|xn*(x)|},(?)x?X.Then LP space X is polyhedral under the new norm.Later,he found a ?unconclitionally convergent series in the space(X,(?))which is not unconditionally convergent,and this implies that every LP space contains a subspace?-isometric to c0 for any ?>0.Motivated by this work,we construct a countable precisely norming set of A,and obtain a basic sequence {xpi} by using the set W,such that(?)fn(xpi)=0,(?)xpi,{fn}(?)W.Then we show that there is an isometrically isomorphic operater between(?)and c0,which proves that LP space contains a subspace isometrically isomorphic to c0.l? is one of the classical sequence spaces,and its dual space can be obtained in two ways.l? is isome trically isomorphic to C(?N),where ?N is the Stone-?ech compactification of the natural nu mbers N.By the Riesz representation theorem,elements in l?*(?)C(?N)*are one-to-one with the fmite countably additive regular signed Borel measures.On the other hand,l?*is isometrically isomorphic to the product space l1?1(c0)° by calculating.However,there are some purely finitely additive measures which coincide with elements in(c0)°,are not countably additive.Therefore,for a countably additive Borel measure,how does it work as a purely finitely additive measure on l??In this paper,we give a general method to get a topological basis of C(?N).Using this hasis,we show that every continuous function in C(?N)is Borel measurable.Beside,for every open set A in ?N,each Borel measure ? restricted to A is finitely additive.Hence,we answer the question above.