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\centerline{\bf The University at Albany}

\centerline{\bf Department of Mathematics and Statistics}

\centerline{\bf Ph.D. Program}

\centerline{\bf Preliminary Examination in Real Analysis}

\centerline{\bf Wednesday, June 3, 1998}

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\item{1.} State the following theorems:

A. Lebesgue dominated convergence theorem.

B. Fatou's lemma.

C. Fubini's theorem.

D. Egoroff's theorem.

E. Radon-Nikodym theorem.

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\item{2.} State and prove from first principles the Lebesgue monotone convergence theorem (in particular do not assume Fatou's lemma.)

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\item{3.} Define what it means for a function, $f$, to be a Lebesgue measurable function.  Prove that the sum of two Lebesgue measurable functions is a Lebesgue measurable function.

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\item{4.} Prove that a real valued continuous function whose domain is a compact subset of the real numbers is uniformly continuous.

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\item{5.} Let $f_n(x)$ be Lebesgue integrable functions on ${\bold{R}}$ such that $\dsize\sum^\infty_{n=1} \dsize\int^\infty_{-\infty} |f_n(x)|dx < \infty$.  Prove that $f_n(x) \to 0$ almost everywhere as $n\to \infty$.

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\item{6.} Let $\{a_{n,k}\} \subset {\bold{R}}$ with $|a_{n,k}|\leq 1$ for $n,k=1,2,3,\dots$ \ .  Suppose that for each $n$, $\dsize\lim_{k\to \infty} a_{n,k}=0$.  Let $p>1$.  Show that
$$\lim_{k\to \infty} \sum^\infty_{n=1} \frac{a_{n,k}}{n^p}=0 \ .$$

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\item{7.} Let $f_n(x)$ be Lebesgue integrable functions on $[0,1]$ such that for each $n=1,2,3,\dots$ $\|f_n\|_1=1$.  Suppose that the Lebesgue measure of the support of $f_n$ tends to zero as $n$ tends to infinity.  Let $p>1$:  Show that $\|f_n\|_p$ tends to infinity as $n$ tends to infinity.

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\item{8.} A.  State the Vitali Covering Theorem.

\itemitem{B.}  Let $f(x)$ be a monotone increasing continuous function on $[0,1]$.  Show that the set of $x$ where $f'(x)=\infty$ is a set of Lebesgue measure zero.  (Hint:  Use part A.)

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\item{9.} Let $f$ and $g$ be non-negative Lebesgue integrable functions on $R$.  Prove that
$$\int^\infty_{-\infty} f(x) \ g(x) \ dx = \int^\infty_0 \varphi(y) \ dy \ ,$$
where $\varphi(y) = \dsize\int_{A_y} g(x) \ dx$ and
$$A_y=\{x: f(x) > y\} \ .$$


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