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=598analysis-missingproblems= | =598analysis-missingproblems= | ||
| + | '''Practice Exam 4''' | ||
| + | 2. Let <math>(X,\mathcal{M}, \mu)</math> be a measure space with <math>\mu(X) =1</math> and let <math>F_1, \ldots, F_{17}</math> be seventeen measurable subsets of <math>X</math> with <math>\mu(F_j)=\frac{1}{4}</math> for every <math>j</math>. | ||
| + | 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 <math>(X,\mathcal{M}, \mu)</math> be a measure space with <math>0<\mu(X) < \infty</math>. Assume that <math>f_n \to f</math> <math>\mu</math>-a.e. and <math>\|f_n\|_p \leq M < \infty</math> for some <math>1<p<\infty</math>. If <math>1\leq r <p</math>, show that <math>f_n \to f</math>/math> in <math>L^r</math>. | |
| − | + | ||
| + | '''Practice Exam 6''' | ||
| + | For <math>n=1,2,\ldots</math>, let <math>f_n:I\to \mathbb{R}, I =[a,b]</math> be a subsequence of functions satisfying the following: If <math>\{x_n\}</math> is a Cauchy sequence in <math>I</math>, then <math>\{f_n(x_n)\}</math> is also a Cauchy sequence. Show that <math>\{f_n\}</math> converges uniformly on <math>I</math>. | ||
[[ 2010 Summer MA 598 Hackney|Back to 2010 Summer MA 598 Hackney]] | [[ 2010 Summer MA 598 Hackney|Back to 2010 Summer MA 598 Hackney]] | ||
Revision as of 04:04, 7 July 2010
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 $.
