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Measure Theory useful for Statistics We shall consider sets of elements or points. The nature of the points needs not be defined, they may represent elementary events, real values, etc. A set is an aggregate of such points. The following operations on two sets A and B are useful: A B , union of A and B , is the set of all points in either A or B A B , intersection of A and B , is the set of all points that belong to both A and B A B , difference of A minus B , is the set of all points in A that are not in B DEFINITION A.1. If is a given set, then a -algebra F on is a family F of subsets of with the following properties: (i) F , where is the empty set (ii) F F FC F , where F \ F is the complement of F in C (iii) A1 , A2 , ... F A: i1 Ai F The pair ( , F ) is called a measurable space. A probability measure P on a measurable space ( , F ) is a function P: F [0,1] such that (a) P( ) 1 (b) 0 P( Ai ) 1 for all sets Ai F i1 i1 (c) if Ai Aj (i j), then P( Ai ) P(Ai ) The triple ( , F , P ) is called a probability space. In a probability context the sets are called events, and we use the interpretation P(F) = "the probability that the event F occurs". 1 An important -algebra used to define random variables is the Borel -algebra on 1 where 1 is the real line. The Borel -algebra on 1 , noted B , is the smallest 1 -algebra containing the collection I of all the open intervals of 1 of the form ( a, b) ={: a < < b}, (-< a b< ) , namely B1 {H : H - algebra of 1 , I H }. It can be shown that the Borel -algebra B does indeed exist. The elements B B are 1 1 called Borel sets, and they include all real intervals; open, closed, semiclosed, finite or infinite. If ( , F , P ) is a given probability space, then a function y: 1 is called F measurable if y 1 (U): { : y( ) U} F . On the strength of the notions introduced we define the random variable as follow DEFINITION A.2: Let ( , F , P ) be a given probability space, and ( 1 , B ) be a 1 measurable space, where B is a -algebra of Borel sets on the real line 1 . A (real 1 valued) random variable x( ) , where are elementary events, is a F measurable mapping from ( , F ) to ( 1 , B ), i.e. x: 1 and 1 B B1 , x 1(B): { : y( ) B} F Every random variable induces a probability measure x on ( , B ), defined by 1 1 2 x (B) P(x 1(B)) P[x x( ) B1 ] From the probability measure x we define the cumulative distribution function of the random variable x Fx ( ) x (I :{' : ' }) P[x ] 3