On a Convexity Condition in Normed Linear Spaces

Abstract

Introduction.The purpose of this paper is to study a convexity property on normed linear spaces (NLS's) which we call P-convexity.Interest in P-convexity is generated by a theorem of Anatole Beck ([1] or [2]) which states that a Banach space 3£ is P-convex if and only if a certain strong law of large numbers is valid for X-valued random variables.Let 3£ be a NLS and (S, 2, m) a measure space.The Borel a-field 38 of X is the tr-field of sets generated by the subsets of BE open in the strong (norm) topology.A Borel set is an element of 38.A function X from 5 into 3t is called strongly measurable if for each Be Sä, X~ \B) = {se S : X(s) e B} e 2. X is called essentially separably valued if there is N eH, m(N) = 0, such that X(S-N) is separable.If js || X(s) || dm{s) < oo and X is essentially separably valued, then X is strongly (Bochner) integrable and there exists ye?), where ?) is the completion of 3£, such that for every x* e £)*, the conjugate space to 9), Js x*X(s) dm(s) = x*y.In this case we write y=js X(s) dm(s)=$s X dm.y is called the strong integral of X.A probability space is a measure space (Í2, 2, 3P) of total measure 1 (^(D)= 1).An 3t-valued random variable on Q. is a function X: Q. -*■ X which is strongly measurable and essentially separably valued.The expectation of X, E{X) = ja X d£? if this integral exists.For a random variable X with expectation, we define the variance of X by (20= f ixw-EvnvtWr). Ja var iWe will call a finite set Xu..., Xn of 3t-valued random variables on Q. independent if for each choice of Borel sets Blt..., Bn e 38, we have d\Xi eB1&-&XneBn) = 3^{XX eB,)---3P{Xn e Bn); that is, n ^{o £ Q : XiUm) e B1 & • ■ & Xn(oj) e Bn) = Y\0>{ojeQ.:X¿w) e Bt}.i = iAn infinite set of random variables is independent if each finite subset is.