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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 Thursday, August 31, 2000}

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\centerline{\bf Part I}

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\item{1.} State the following theorems:

A. The Lebesgue Dominated 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.}  Given the properties of measure theory, give a definition of the Lebesgue integral.

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\centerline{\bf Part II}

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

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\item{3.} Prove if $E$ and $F$ are subsets of the real line with positive Lebesgue measure and, if
$$E+F=\{x: x=y+z \ \text{with} \ y\in E \ \text{and} \ z\in F\}$$
then $E+F$ contains a non-empty open interval.

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\item{4.} A. Construct the Cantor ternary set, $C$.

B. Prove that $C$ has Lebesgue measure 0.

C. Prove that the characteristic function of $C$ is Riemann integrable.

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\item{5.} Prove there exists  a non-measurable subset of the real line.

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\item{6.} Let $E\subseteq [0,1)$ where $x\in E$ if $x$ has no 9's in its decimal expansion.  Prove that $E$ has Lebesgue measure 0.

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\item{7.} If $f$ is a measurable function on $[a,b]$, $a<b$ and we define
$$\|f\|_\infty = \inf\{M\geq 0: |E_M|=0\}$$
where $E_M=\{x: |f(x)|>M\}$.  $\|f\|_\infty$ is called the essential supremum of $f$.

Prove that $\|f\|_\infty = \dsize\lim_{p\to \infty} \|f\|_p = \dsize\lim_{p\to \infty} \left[\dsize\int^b_a |f(x)|^p dx\right]^{1/p}$.

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\item{8.} Prove that the infinite sum
$$\sum^\infty_{n=0} \int^{\pi/3}_0 (1- \sqrt{\sin x})^n \cos x \ dx$$
has a finite limit and find its value.

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\item{9.} Prove if $f_n$ is a sequence of Lebesgue integrable functions with $\|f_n\|{_1}=1$ for $n=1,2,3,\dots$ and if the measure \{support of $f_n$\} $\to 0$ as $n\to \infty$ then for all $p>1$ we have
$$\|f_n\|{_p} \to \infty \ \ \text{as} \ n\to \infty \ .$$

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\item{10.} Prove that if $f$ is a differentiable function on $(-\infty, \infty)$ such that $f$ and $f'$ are both in $L^1(-\infty, \infty)$ then
$$\int^\infty_{-\infty} f'(x) dx = 0 \ .$$


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