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\centerline{\bf{Ph.D. Preliminary Exam in Real Analysis}}

\centerline{\bf{August 1994}}

Do problems 1 and 2 and as many of the remaining as time permits.  Strive for complete solutions.

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\item{1.} State the following:

(a) Fatou's Lemma

(b) Monotone Convergence Theorem

(c) Lebesgue Dominated Convergence Theorem

(d) H\"olders Inequality

(e) Egoroff's Theorem

(f) Fubini's Theorem

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\item{2.} Starting with basic principles, prove 1.(a), (b) and (c) in any order.

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\item{3.} (a) Prove that the sum of two Lebesgue measurable functions is measurable.

\item\item{(b)} Give an example of a function which is in $L^3([0,1])$ but not in $L^2([0,1])$ (Lebesgue) and justify your answer.

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\item{4.} Evaluate $\dsize\lim_{n\to \infty} \ \dsize\int^{\pi/2}_0 ne^{-x^2} \cos x \sin^{n-1}x \ dx$, justifying your computations.

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\item{5.} Let $(X,{\cal{B}},\mu)$ be a $\sigma$-finite measure space and let $\nu$ be a measure such that $\nu<<\mu$

\item\item{(a)} State the Radon-Nikodym Theorem in this context.

\item\item{(b)} If $f$ is a nonnegative measurable function, show that $\dsize\int fd\nu=\dsize\int f[\dsize\frac{d\nu}{d\mu}]d\mu$ where [ ] denotes the $R-N$ derivative.

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\item{6.} (a) Let $f: [0,1]\to {\bold{R}}$ be given by $f(0)=0$, $f(x)=x^{1/2}$ for $x>0$.  Show by direct computation that $f$ is absolutely continuous.

\item\item{(b)} Let $g: [0,1]\to {\bold{R}}$ be given by $g(0)=0$, $g(x)=x \sin \ \dsize\frac{1}{x}$ for $x>0$.  Is $g$ of bounded variation?  (Justify completely.)

\item\item{(c)} Is every continuous function on $[0,1]$ absolutely continuous?  (Justify in reasonable detail.)

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\item{7.} Let $\mu$ = Lebesgue measure on $[0,1]$ and recall that, for a measurable function $f$, $\|f\|_\infty$ is defined to be inf$\{m: \mu\{t\big| |f(t)|>m\}=0\}$.  Prove that if $\|f\|_\infty<\infty$, then $f$ is in $L^p$ for all $p\geq 1$ and that $\dsize\lim_{p\to \infty}\|f\|_p=\|f\|_\infty$.

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\item{8.} Let $X=Y=[0,1]$ and $\mu=\nu$ = Lebesgue measure.

\item\item{(a)} Prove that each Borel set in $X\times Y$ is measurable with respect to $\mu\times \nu$.

\item\item{(b)} Let $h$ and $g$ be $\mu$-integrable on [0,1] and define $f$ on $X\times Y$ by $f(x,y)=h(x)\cdot g(y)$.  Prove that $f$ is integrable and establish a formula for $\dsize\int_{X\times Y} fd(\mu\times \nu)$ in terms of $h$ and $g$.

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\item{9.} (a) Show that the (improper) Riemann integral of a function may exist, on say $[0,\infty)$, although the function may not be Lebesgue integrable on $[0,\infty)$.

\item\item{(b)} Let $f\geq 0$ be Lebesgue integrable on ${\bold{R}}$.  Show that the function $F(x)=\dsize\int^x_{-\infty} f$ is continuous.

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