598analysis-missingproblems
Practice Exam 4
2. Let $ (X,\mathcal{M}, \mu) $ be a measure space with $ \mu(X) =1 $ and let $ F_1, \ldots, F_{17} $ be seventeen measurable subsets of $ X $ with $ \mu(F_j)=\frac{1}{4} $ for every $ j $. a. Prove that (some) five of these subsets must have an intersection of positive measure. b. Is the conclusion above true if we take sixteen sets instead of seventeen?
5. Let $ (X,\mathcal{M}, \mu) $ be a measure space with $ 0<\mu(X) < \infty $. Assume that $ f_n \to f $ $ \mu $-a.e. and $ \|f_n\|_p \leq M < \infty $ for some $ 1<p<\infty $. If $ 1\leq r <p $, show that $ f_n \to f $/math> in $ L^r $.
Practice Exam 6
For $ n=1,2,\ldots $, let $ f_n:I\to \mathbb{R}, I =[a,b] $ be a subsequence of functions satisfying the following: If $ \{x_n\} $ is a Cauchy sequence in $ I $, then $ \{f_n(x_n)\} $ is also a Cauchy sequence. Show that $ \{f_n\} $ converges uniformly on $ I $.
