245B_W09_hwk1

245B_W09_hwk1 - 245B, Winter 2009, Assignment 1: Notes and...

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245B, Winter 2009, Assignment 1: Notes and selected model answers (Model solutions follow question numbers in bold .) Folland Chapter 3 3. a. Our convention for a signed measure ν is simply to deﬁne L 1 ( ν ) := L 1 ( ν + ) L 1 ( ν - ) in terms of the Jordan Decomposition, and now f L 1 ( ν + ) L 1 ( ν - ) Z | f | d ν + < Z | f | d ν - < Z | f | d ν + + Z | f | d ν - < ( both values 0 ) Z | f | d | ν | < f L 1 ( | ν | ) . [Note: although it looks tempting, one can’t really ‘deduce’ from ± ± R | f | d ν ± ± < that R | f | d ν + < and R | f | d ν - < , because by deﬁnition for a signed measure we only allow ourselves to write ‘ R | f | d ν ’ at all given these two separate ﬁniteness conditions.] b. It sufﬁces for f L 1 ( ν ) = L 1 ( | ν | ) to write in terms of the Jordan Decom- position ν = ν + - ν - that ± ± ± Z f d ν ± ± ± = ± ± ± Z f d ν + - Z f d ν - ± ± ± ± ± ± Z f d ν + ± ± ± + ± ± ± Z f d ν - ± ± ± Z | f | d ν + + Z | f | d ν - = def Z | f | d | ν | . c. On the one hand, if X = X + X - is a Hahn Decomposition for ν , then 1

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f := χ X + - χ X - satisﬁes | f | ≤ 1 and ± ± ± Z E f d ν ± ± ± = ν ( E X + ) - ν ( E X - ) = def ν + ( E ) + ν - ( E ) = | ν | ( E ) , so the supremum over all such f is certainly at least | ν | ( E ) . On the other hand, for any measurable f with | f | ≤ 1 part (b) and monotonicity for the (unsigned) measure | ν | give ± ± ± Z E f d ν ± ± ± Z E | f | d | ν | ≤ Z E χ X d | ν | = | ν | ( E ) , giving the reverse inequality. 7. Be careful for both parts (as in all other evaluations of a supremum) to prove carefully that (a) a claimed ‘supremum’ is an upper bound for the desired set of values, and (b) that no smaller number is an upper bound for that set of values, or, equivalently, that that set of values comes arbitrarily close to the ‘supremum’ from below; and similarly for inﬁma. It turns out in these cases that you can achieve equality for some member of the set of values. 10. Whichever option you choose, be careful to prove both (a) that ν ± μ (i.e., μ ( E ) = 0 ν ( E ) = 0 ) and (b) that for some ε > 0 , for every δ > 0 there is a measurable set E with μ ( E ) < δ but ν ( E ) ε . 11.
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This note was uploaded on 04/06/2010 for the course MATH various taught by Professor Tao/analysis during the Spring '10 term at UCLA.

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245B_W09_hwk1 - 245B, Winter 2009, Assignment 1: Notes and...

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