Mathematics 105 — Spring 2004 — M. ChristProblem Set 9For 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 Theorem3.4.8 and the related inequality (3.4.1).Theorem 3.4.8 expresses a nonobvious and fundamentalproperty of integrable functions. Please read§3.4 in this spirit; you are very welcome to study theproof of (3.4.7) and the statement and proof of the rising sun (aka sunrise) lemma, but these arenot 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 finitemeasure space Professor Stroock means thatμ(E)<∞, not thatEis a finite set.The symbol“x∧y” means “min(x, y)”, the minimum ofx, y.) Solve the following problems.IX.AConsider the measure space (R1,BR1, λ) whereλdenotes Lebesgue measure. Consider themeasurable functionsfn(x
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measure, Lebesgue measure, Lebesgue integration, measure space