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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 Friday, June 11, 1999}

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\centerline{\bf PART I}

\centerline{\bf Do problems 1 and 2:}

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\item{1.} State the following theorems:

A. The Lebesgue Monotone Convergence Theorem,

B. Fatou's Lemma,

C. Egoroff's Theorem,

D. The Fubini Theorem,

E. H\"older's Inequality.

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\item{2.} A. Give a definition of the Lebesgue integral.

\itemitem{B.} Sketch a proof of the Lebesgue Monotone Convergence Theorem assuming only the basic properties of measure theory.

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\centerline{\bf PART II}

\centerline{\bf Do 6 of the following 8 problems:}

\item{3.} If $f$ and $g$ are Lebesgue measurable functions on $I=[0,1]$ then $f(x)-g(x)$ is also a Lebesgue measurable function.

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\item{4.} Let $\{f_n\}$ be a sequence of non-negative Lebesgue measurable function is on $I=[0,1]$ with
$$f_1(x) \geq f_2(x) \geq f_3(x) \dots \geq f_n(x) \geq \dots \geq 0$$
for each $x\in I$ and assume $\dsize\lim_{n\to \infty} \dsize\int^1_0 f_n(x) dx = 0$.  Prove that for almost every $x\in I$,
$$\lim_{n\to \infty} f_n(x) = 0 \ .$$

\item{5.} Does there exist a strictly increasing function defined on an interval I so that $f'(x)=0$ almost everywhere on $I$?  Prove your answer.

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\item{6.} A. Let $\mu$ and $\nu$ be Borel measures on $[0,1]$.  Define the Radon-Nikodym derivative of $\nu$ with respect to $\mu$.

\itemitem{B.} Let $\lambda$ and $\mu$ be Borel measures on $[0,1]$.  Show that $\mu$ is absolutely continuous with respect to $\lambda + \mu$.

\itemitem{C.} Let $\lambda$ and $\mu$ be Borel measures on $[0,1]$.  Show that
$$\frac{d\lambda}{d(\lambda+\mu)} + \frac{d\mu}{d(\lambda+\mu)} = 1, \ (\lambda+\mu) \ \text{almost everywhere} \ \ .$$

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\item{7.} Suppose $f$ is Lebesgue integrable on $R^+$ and let
$$g(x) = \int^\infty_0 \ \frac{f(t)}{x+t} \ dt, \ \ x>0 \ .$$
Is $g$ continuous?  Does $g$ have a limit at $x\to \infty$?  Is $g$ differentiable?  Prove your answers.

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\item{8.} Let $\{f_n\}$ be a sequence of continuous functions on $I=[0,1]$.  Prove that the set of points where $\dsize\lim_{n\to \infty} f_n(x)=0$ is an $F_{\sigma \delta}$ Borel set.

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\item{9.} Let $\{E_n\}$ be a sequence of Lebesgue measurable subsets of $[0,1]$ such that for each $n$, $|E_n|\geq \delta >0$.  Suppose that $c_n$ is a sequence of non-negative real numbers such that $\dsize\sum^\infty_{n=1} c_n \chi_{_{E_n}}(x)< \infty$ for almost every $x\in [0,1]$.  Show that $\dsize\sum^\infty_{n=1} c_n < \infty$.

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\item{10.} If $\{f_n\}$ is a sequence of measurable functions on $I=[0,1]$ with $\dsize\int^1_0 |f_n(x)|^2 dx \leq 1$ for each $f_n$, and if $\dsize\lim_{n\to \infty} f_n(x) = f(x)$ for almost every $x\in I$ prove that
$$\lim_{n\to \infty} \int^1_0 |f_n(x) - f(x)|dx = 0 \ .$$
(Hint:  Use Fatou's lemma and Egoroff's theorem.)

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