Table of contents

Real Analysis I practice problems

Question 0.1.
Let (X, \cM, \mu) be a finite measure space. Suppose X admits a sequence of finite partitions \cF_n consisting of measurable sets, such that \sup \{\mu(F) \st F \in \cF_n\} \to 0 as n \to \infty. Prove that for any t \in [0, \mu(X)], there exists A \in \cM such that \mu(A) = t.
(We say (X, \cM, \mu) has the intermediate value property.)
Question 0.2.
Suppose \mu is a finite measure on (X, \cM) that is nonatomic, that is, for A \in \cM and \mu(A) > 0, there exists B \in \cM and B \subset A such that 0 < \mu(B) < \mu(A).
  1. Prove that for every \epsilon > 0 and A \in \cM, there exists a measurable set C \subset A such that 0 < \mu(C) \le \epsilon.
  2. Prove that for every \epsilon> 0, there exists a finite partition of X into measurable subsets of measure at most \epsilon.
    (Hint: Define \alpha_1 = \sup\{\mu(E) \st E \subset X, \, \mu(E) \le \epsilon\} > 0, and let B_1 \subset X be such that \alpha_1/2 < \mu(B_1) \le \alpha_1. Considering X_1 = X \setminus B_1 and continue inductively unless \mu(X_n) = 0.)
  3. Conclude that the measure space has the intermediate value property as formulated in the previous question.
    (For this reason, nonatomic measures are sometimes called continuous.)
Question 0.3.
Let X be a metric space. An outer measure on \cP(X) is called a metric outer measure if for all sets A, B such that d(A, B) = \inf \{d(x, y) \st x \in A, \, y \in B\} > 0, we have
\mu^*(A \cup B) = \mu^*(A) + \mu^*(B).
Prove that if \mu^* is a metric outer measure, then all open sets are measurable.
Question 0.4.
Let R_\alpha: [0, 1) \to [0, 1) be given by R_\alpha(x) = x + \alpha \mod 1. Show that if \alpha \notin \Q, then \cO_\alpha(x) = \{x + n\alpha \st n \in \Z\} is dense in [0, 1).