No2(2003)solutions

No2(2003)solutions - MATH 587 Solutions to Assignment 2(1 f...

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MATH 587 Solutions to Assignment 2 October 23, 2003 (1) f 1 ( σ ( C )) is a σ -algebra containing f 1 ( C ), so contains σ ( f 1 ( C )). Let B = { B 0 : f 1 ( B ) σ [ f 1 ( C )] } . Then B is a σ -algebra containing C , so contains σ ( C ). Hence f 1 ( σ ( C )) σ ( f 1 ( C )). (2) (a) Since ( a, b ) = lim n →∞ ( a, b n ], where a<b n <b n +1 <b and b n b , then µ ( a, b )= lim n →∞ µ ( a, b n ] = lim n →∞ [ F ( b n ) F ( a )] = F ( b ) F ( a ). (b) Since [ a, b ] = lim n →∞ ( a n ,b ], where a n <a n +1 <a and a n a , then µ [ a, b ] = lim n →∞ µ ( a n ,b ]= lim n →∞ [ F ( b ) F ( a n )] = F ( b ) F ( a ). (c) Since [ a, b ) = lim n →∞ ( a n ,b ), where a n <a n +1 <a and a n a , then µ [ a, b ) = lim n →∞ µ ( a n ,b )= lim n →∞ [ F ( b ) F ( a n )] = F ( b ) F ( a ). (d) Since
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