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2004 Apr

2004 April 19

Instructions. Use a separate sheet of paper for each new problem. Do as many problems as you can. Complete solutions to five problems will be considered as an excellent performance. Be advised that a few complete and well written solutions will count more than several partial solutions.\ Notation. \(f\in C(X)\) means that \(f\) is a real-valued, continuous function defined on \(X\).

  1. (a) Let \(S\) be a (Lebesgue) measurable subset of \(\mathbb R\) and let \(f, g: S\to \mathbb R\) be measurable functions. Prove:\ i. \(f+g\) is measurable and\ ii. if \(\phi \in C(\mathbb R)\), then \(\phi(f)\) is measurable.\ (b) Let \(f\colon [a,b]\to [-\infty, \infty]\) be a measurable function. Suppose that \(f\) takes the value \(\pm \infty\) only on a set of (Lebesgue) measure zero. Prove that for each \(\epsilon>0\) there is a positive number \(M\) such that \(|f|\leq M\), except on a set of measure less than \(\epsilon\).

  2. (a) State Egorov's theorem.\ (b) State Fatou's lemma.\ (c) Let \(\{f_n\} \subset L^p[0,1]\), where \(1\leq p< \infty\). Suppose that \(f_n\to f\) a.e., where \(f\in L^p[0,1]\). Prove that \(\\|f_n - f\\|_p\to 0\) if and only if \(\\|f_n\\|_p\to \\|f\\|_p\).

  3. (a) Let \(S = [0, 1]\) and let \(\{f_n\} \subset L^p(S)\), where \(1< p< \infty\). Suppose that \(f_n\to f\) a.e. on \(S\), where \(f\in L^p(S)\). If there is a constant \(M\) such that \(\|f_n\|_p\leq M\) for all \(n\), prove that for each \(g\in L^q(S), \frac{1}{p} + \frac{1}{q} = 1\), we have \(\lim\limits_{n\to \infty} \int_S f_n g = \int_S fg\).\ (b) Show by means of an example that this result is false for \(p=1\).

  4. State and prove the closed graph theorem.

  5. Prove or disprove:\ (a) For \(1 \leq p < \infty\), let \(\ell^p = \left\{ \mathbf{x} = \{x_k\} \bigm\vert \\| \mathbf x \\|_p = \left(\sum_{k=1}^\infty|x_k|^p\right)^{1/p} < \infty \right\}\). Then for \(p \neq 2\), \(\ell^p\) is a Hilbert space.\ (b) Let \(X = (C[0,1], \\| \cdot \\|_1)\), where the linear space \(C[0,1]\) is endowed with the \(L^1\)-norm: \(\\|f\\|_1 = \int_0^1 |f(x)|\ dx\). Then \(X\) is a Banach space.\ (c) Every real, separable Hilbert space is isometrically isomorphic to \(\ell^2\).

  6. Let \(f, g \in L^1(\mathbb R)\). Give a precise statement of some version of Fubini's theorem that is valid for non-negative functions, and then prove the following:\ (a) \(h(x) = \int_\mathbb R f(x-t)g(t)\ dt\) exists for almost all \(x\in \mathbb R\);\ (b) \(h\in L^1(\mathbb R)\) and \(\\|h\\|_1 \leq \\|f\\|_1 \\|g\\|_1\).

  7. (a) State the Radon-Nikodym theorem.\ (b) Let \((X, \mathcal B, \mu)\) be a complete measure space, where \(\mu\) is a positive measure defined on \(\mathcal B\), a \(\sigma\)-algebra of subsets of \(X\). Suppose \(\mu(X) < \infty\) and \(S\) is a closed subset of \(\mathbb R\). Let \(f\in L^1(\mu)\), where \(f\) is an extended real-valued function defined on \(X\). Prove: If for every \(E\in \mathcal B\) with \(\mu(E) > 0\) we have

    \[A_E(f) = \frac{1}{\mu(E)}\int_E f\ d\mu \in S,\]

    then \(f(x)\in S\) for almost all \(x\in X\).