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# hw9 - Mathematics 105 Spring 2004 M Christ Problem Set 9...

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Mathematics 105 — Spring 2004 — M. Christ Problem Set 9 For Friday April 16: Continue to study § 3.3 of our text. We will treat § 3.4 in a somewhat super- ficial way by discussing the statement (3.4.7) but not its proof , and showing how it implies Theorem 3.4.8 and the related inequality (3.4.1). Theorem 3.4.8 expresses a nonobvious and fundamental property of integrable functions. Please read § 3.4 in this spirit; you are very welcome to study the proof of (3.4.7) and the statement and proof of the rising sun (aka sunrise) lemma, but these are not officially part of this course and you will not be examined on them. After this abbreviated discussion of § 3.4 we will move on to § 4.1 and then § 5.1. Solve the following problems from Stroock § 3.3: 3.3.21 parts (i),(ii),(iii). 3.3.22 (By a finite measure space Professor Stroock means that μ ( E ) < , not that E is a finite set. The symbol x y ” means “min( x, y )”, the minimum of x, y .) Solve the following problems. IX.A Consider the measure space ( R 1 , B R 1 , λ ) where λ denotes Lebesgue measure. Consider the measurable functions f n ( x
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