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2007 Nov

2007 November 16

Notation. \(\mathbb R\) is the set of real numbers and \(\mathbb R^n\) is \(n\)-dimensional Euclidean space. Denote by \(m\) Lebesgue measure on \(\mathbb R\) and \(m_n\) \(n\)-dimensional Lebesgue measure. Be sure to give a complete statement of any theorems from analysis that you use in your proofs below.

  1. Let \(\mu\) be a positive measure on a measure space \(X\). Assume that \(E_1, E_2, \dots\) are measurable subsets of \(X\) with the property that for \(n\neq m, \mu(E_n\cap E_m) = 0\). Let \(E\) be the union of these sets. Prove that \(\mu(E) = \sum_{n=1}^\infty \mu(E_n)\).

  2. (a) State a theorem that illustrates Littlewood's Principle for pointwise a.e. convergence of a sequence of functions on \(\mathbb R\).\ (b) Suppose that \(f_n \in L^1(m)\) for \(n=1,2,\dots\). Assuming that \(\\|f_n-f\\|_1 \to 0\) and \(f_n \to g\) a.e. as \(n\to \infty\), what relation exists between \(f\) and \(g\)? Make a conjecture and then prove it using the statement in Part (a).

  1. Let \(K\) be a compact subset in \(\mathbb R^3\) and let \(f(x) = \mbox{dist}(x,K)\).\ (a) Prove that \(f\) is a continuous function and that \(f(x) = 0\) if and only if \(x\in K\).\ (b) Let \(g = \max \{1-f, 0\}\) and prove that \(\lim_{n\to \infty} \iiint g^n\) exists and is equal to \(m_3(K)\).

  2. Let \(E\) be a Borel subset of \(\mathbb R^2\).\ (a) Explain what this means.\ (b) Suppose that for every real number \(t\) the set \(E_t = \{(x,y) \in E \mid x=t\}\) is finite. Prove that \(E\) is a Lebesgue null set.

  3. Let \(\mu\) and \(\nu\) be finite positive measures on the measurable space \((X,\mathcal A)\) such that \(\nu \ll \mu \ll \nu\), and let \(d\nu/d(\mu + \nu)\) denote the Radon-Nikodym derivative of \(\nu\) with respect to \(\mu+\nu\). Show that

    \[0 < \frac{d\nu}{d(\mu + \nu)} < 1 \quad \text{a.e.} [\mu].\]

  4. Suppose that \(1 < p < \infty\) and \(q = p/(p-1)\) and let \(a_1, a_2, \dots\) be a sequence of real numbers for which the series \(\sum a_n b_n\) converges for all real sequences \(\{b_n\}\) satisfying the condition \(\sum |b_n|^q < \infty\).\ (a) Prove that \(\sum |a_n|^p < \infty\).\ (b) Discuss the cases of \(p=1\) and \(p=\infty\). Prove your assertions.