Exam
Ph.D. Preliminary Examination in Real Analysis
Spring 2022 · 6 questions in this exam.
Questions in This Exam
Ordered by the question sequence in the original exam.
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Question 1
Prove the Borel-Cantelli lemma: Let \((X, \mathcal {M}, \mu )\) be a measure space and let \(A_{i}\) be measurable sets such that \(\sum _{i=1}^{\infty }\mu (A_{i}) < \infty \). Then the set
\[ \Omega = \{x \in X \mid x \in A_{i}\text { for infinitely many } i\} \]of points that belong to infinitely many \(A_{i}\) has measure 0.
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Question 2
Let \(X\) be a compact metric space, \(\mathcal {M}\) the Borel \(\sigma \)-algebra on \(X\), and \(\mu \) a finite measure on \(\mathcal {M}\) (meaning that \(\mu (X) < \infty \); we call such a measure a finite Borel measure on \(X\)). Prove that \(\mu \) is regular, i.e. for every Borel set \(E \subset X\):
- \(\mu (E) = \sup \{\mu (K) \mid K \subseteq E \text { is compact}\}\), and
- \(\mu (E) = \inf \{\mu (U) \mid U \supseteq E \text { is open}\}\).
Hint: Show that the set of \(E\)’s that satisfy both bullets is a \(\sigma \)-algebra that contains compact sets.
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Question 3
- Let \((X, \mathcal {M})\) be a measurable space, \(\mu , \nu \) positive measures on \((X, \mathcal {M})\). Suppose \(\nu (X) < \infty \). Prove that \(\nu \) \(\ll \mu \) if and only if for every \(\epsilon > 0\) there exists \(\delta > 0\) such that \(\mu (E) < \delta \) implies \(\nu (E) < \epsilon \).
- For \(\mu \) the Lebesgue measure on \(\mathbb {R}\) find a positive (but infinite) measure \(\nu \) such that \(\nu \) \(\ll \mu \) but the \(\epsilon \)-\(\delta \) statement in (a) fails.
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Question 4
Let \((X, \mathcal {M}, \mu )\) be a measure space with \(\mu (X) < \infty \). Let \(f : X \to X\) be a measure preserving bijection, in the sense that \(\mu (A) = \mu (f(A))\) for every \(A \in \mathcal {M}\). Prove that for every \(E \in \mathcal {M}\)
\[ \{x \in E \mid f^{n}(x) \notin E \text { for all } n > 0\} \]has measure 0.
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Question 5
Let \(V\), \(W\) be Banach spaces, and let \(T_{i} : V \to W\) be a sequence of bounded linear operators such that \(\lim _{i \to \infty } T_{i}v\) exists for every \(v \in V\). Define the linear map \(T : V \to W\) by \[ Tv = \lim _{i \to \infty } T_{i}v. \]
Show that \(T\) is bounded.
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Question 6
Show that the Banach space \(\ell ^{p}\) with the usual norm is not a Hilbert space
when \(p \in [1, 2) \cup (2, \infty )\).