Exam
Ph.D. Preliminary Examination in Real Analysis
Fall 2021 · 6 questions in this exam.
Questions in This Exam
Ordered by the question sequence in the original exam.
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Question 1
Let \((X, \mathcal {M})\) be a measurable space, so that \(\mathcal {M}\) is a \(\sigma \)-algebra of subsets of \(X\). Prove that \(\mathcal {M}\) is either finite or uncountable.
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Question 2
Let \((X, \mathcal {M}, \mu )\) be a measure space with \(\mu (X) = 1\) and let \(E_{1}, \ldots , E_{n}\) be measurable subsets of \(X\). Suppose that every \(x \in X\) belongs to at least \(k\) of the sets \(E_{i}\). Show that at least one of the sets \(E_{i}\) has measure
\[ \mu (E_{i}) \ge \frac {k}{n}. \] -
Question 3
Show that there does not exist a Lebesgue measurable subset \(A \subset \mathbb {R}\) such that for every interval \((a, b) \subset \mathbb {R}\) we have \[ m(A \cap (a, b)) = \frac {b-a}{2} \] where \(m\) denotes Lebesgue measure.
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Question 4
Let \(f : [0, 1] \to \mathbb {R}\) be a continuous function and define a signed measure \(\mu \) on the Lebesgue measurable subsets of \([0, 1]\) by
\[ \mu (E) = \int _{E} f\,dm \]where \(m\) is Lebesgue measure. Suppose that for all \(n = 0, 1, 2, 3, \ldots \)
\[ \int _{[0,1]} x^{n}\,d\mu = 0 \]Prove that \(\mu = 0\).
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Question 5
Let \(f, g \in L^{2}(\mathbb {R})\). Show that
\[ h(x) = \int _{\mathbb {R}} f(x-y)g(y)\,dy \]is a well-defined real number for every \(x \in \mathbb {R}\) and that \(h : \mathbb {R} \to \mathbb {R}\) is continuous.
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Question 6
Prove or disprove:
- \(\ell ^{1}\) is separable.
- \(\ell ^{\infty }\) is separable.