Real Analysis Exam

A part of the phd exam in analysis

Reading time: about 17 minutes (3519 words).

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 $\mu(E) = \sum_{n=1}^\infty \mu(E_n)$.
  1. (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 $$0 < \frac{d\nu}{d(\mu + \nu)} < 1 \quad \text{a.e.} [\mu].$$

  1. 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.