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Measure Theory Notes

Measure Theory Notes - Economics 245A Notes for Measure...

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Economics 245A Notes for Measure Theory Lecture Axiomatic Approach The axiomatic approach introduced by Kolmogorov starts with a set of axioms, as do all axiomatic approaches, that are taken to be obvious. The axioms must be: (1) complete, that is all theorems can be proven from the axioms; (2) consistent, that is the axioms do not contradict each other; and (3) non-redundant, that is no axiom can be derived from the other axioms. Technical Detail for Probability Axiom (iii) The statement of probability axiom (iii) that appears in most textbooks is: If A 1 ; : : : ; A n are disjoint sets then P ( [ n i =1 A i ) = n X i =1 P ( A i ) : (0.1) Probability axiom (iii) given in class refers to the quantity [ 1 i =1 A i , so we must show that (0.1) continues to hold for n = 1 . To verify, let A = [ 1 i =1 A i , B n = [ n i =1 A i , and let C n = A ° B n : Then f C n g 1 n =1 is a monotone decreasing sequence and lim n !1 C n = \ 1 n =1 C n = ; : For any value of n : P ( A ) = n X i =1 P ( A i ) + P ( C n ) ; so we must prove that P ( C n ) ! 0 as n ! 1 . Proof: Method, proof by contradiction. Assume that there exists some ° > 0 such that P ( C n ) > ° for all n . Let v ( s ) be the largest value of n for which s 2 C n : Then P ( f s : v ( s ) > n g ) = P ( C n +1 ) ; because C n +1 consists of all events for which v ( s ) > n . Yet any element s of the sample space belongs to only °nitely many C n because \ 1 n =1 C n = ; . Because any element belongs to only °nitely many C n it is not possible for P ( C n ) > ° for all n . Hence P ( C n ) ! 0 as n ! 1 .
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