# Problem 1. (a) Prove the following relative of the Markov-Chebyshe...

## Transcribed Text

Problem 1. (a) Prove the following relative of the Markov-Chebyshev inequality: Let f : R R be Lebesgue measurable and let O > 0. Then (Here f2 is the function f2(x) = f(x)2) (b) Let fn : R R, n = 1,2,..., be a sequence of Lebegue measurable functions. Assume that mj fn < 80. Using part (a), prove that for every € > 0 we have n (c) Use part (b) and the Borel-Cantelli lemma to conclude that fn 0 pointwise a.e. [Note: the conclusion that fn 0 pointwise a.e. can also be deduced from the monotone convergence theorem (MCT); this problem outlines a proof that avoids directly applying MCT.] Problem 2. Recall: A subset E of R is called nowhere dense if its closure has empty interior (i.e., the complement of the closure of E is open and dense in R). A subset E of R is called meager if it is contained in a countable union of nowhere dense sets. (a) Prove that for every € > 0 there exists a closed, nowhere dense subset C of R with m(R - C) (b) Prove that there exists a meager subset M of R with m(R - M) (c) Prove or disprove: there exists a nowhere dense subset C of R with m (IR - C) = 0. Problem 3. Let (fn) and (gn) be sequences in L¹(R) and let g € L' (R). Assume that (i) Ifnl < In a.e. for all 72 (ii) In converges pointwise a.e. to g (iii) S 9n J`g (iv) fn converges pointwise a.e. to a function f. Prove that f € L' (R) and that j fn J'f. [Note and Hint: The special case where In = g for all n recovers the Dominated Conver- gence Theorem. To prove this, follow all the steps in the proof the Dominated Convergence Theorem, making any necessary changes along with way.]

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