σ-Fields and Probability.pdf

1 σ-Fields and Probability Andrzej Ne?dzusiak Warsaw University Bia?ystok Summary. This article contains de?nitions and theorems concerning basic properties of following objects: - a ?eld of subsets of given nonempty set; - a sequence of subsets of given nonempty set; - aσ-?eld of subsets of given nonempty set and events fromthisσ-?eld; - a probability -additive normed measure de?ned on previously introducedσ-?eld; - a σ-?eld generated by family of subsets of given set; - family ofBorel Sets. MML Identi?er: PROB_1. WWW: /JFM/Vol1/ The articles [8], [4], [11], [10], [12], [2], [3], [1], [9], [6], [5], and [7] provide the notation and terminology for this paper. For simplicity, we use the following convention:O 1denotes a non empty set,X,Y,Z,p,x,y, zdenote sets,Ddenotes a subset ofO 1,fdenotes a function,m,ndenote natural numbers,r,r 2 denote real numbers, ands 1denotes a sequence of real numbers. The following two propositions are true: (2) 1For allr,r 2such that 0≤rholdsr 2?r≤r 2. (3) For allr,s 1such that there existsnsuch that for everymsuch thatn≤mholdss 1(m) =r holdss 1is convergent and lims 1=r. LetXbe a set and letI 1be a family of subsets ofX. We say thatI 1is closed plement operator if and only if: (Def. 1) For every subsetAofXsuch thatA∈I 1holdsA c∈I 1. LetXbe a set. One can verify that there exists a family of subsets ofXwhich is non empty, closed ple