hw7 - Math 245A Homework 7 Brett Hemenway March 25, 2007...

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Unformatted text preview: Math 245A Homework 7 Brett Hemenway March 25, 2007 Folland Chapter 3 1. Let be a signed measure on ( X, M ). a. If { E j } is an increasing sequence in M , if we set E = , then the sets E j \ E j- 1 are disjoint, so [ j =1 E j = X j =1 ( E j \ E j- 1 ) = lim n n X j =1 ( E j \ E j- 1 ) = lim n n X j =1 ( E j )- ( E j- 1 ) = lim n ( E n ) b. If { E j } is a decreasing sequence in M , and | ( E 1 ) | < , then Let F j = E 1 \ E j . Then F i F i +1 , ( E 1 ) = ( F j ) + ( E j ), and [ j =1 F j = E 1 \ j =1 E j 1 By (a) we have ( E 1 ) = j =1 E j + [ j =1 ( F j ) = j =1 E j + lim j ( F j ) = j =1 E j + lim j [ ( E 1 )- ( E j )] = j =1 E j - ( E 1 ) + lim j ( E j ) Subtracting ( E 1 ) from both sides gives j =1 E j = lim j ( E j ) 4. Let be a signed measure on ( X, M ), and , positive measures such that = - . Let E,F such that E F = X , E F = , and + ( F ) = - ( E ) = 0. For A E , B F , we have ( A )- ( A ) = ( A ) = + ( A ) So ( A ) + ( A ) since is positive, and ( B ) 0 = + ( B ). Simi- larly, ( B )- ( B ) = ( B ) =- - ( B ) So...
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hw7 - Math 245A Homework 7 Brett Hemenway March 25, 2007...

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