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 εn>0,ε1<1.εn↓0,define(?){(1+εn)|xn*(x)|},(?)x∈X.Then LP space X is polyhedral under the new norm.Later,he found a ωunconclitionally convergent series in the space(X,(?))which is not unconditionally convergent,and this implies that every LP space contains a subspaceε-isometric to c0 for any ε>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(βN),where βN is the Stone-?ech compactification of the natural nu mbers N.By the Riesz representation theorem,elements in l∞*(?)C(βN)*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(βN).Using this hasis,we show that every continuous function in C(βN)is Borel measurable.Beside,for every open set A in βN,each Borel measure μ restricted to A is finitely additive.Hence,we answer the question above.