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\centerline{\bf University at Albany}

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\centerline{\bf Department of Mathematics and Statistics}

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\centerline{\bf Preliminary Examination}

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\centerline{\bf Real Analysis}

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\centerline{\bf August, 1998}

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\centerline{\ }

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Do as many problems as possible!

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1. State the following theorems.

a. Monotone Convergence Theorem

b. Fatou's Lemma

c. Dominated Convergence Theorem

d. H\"older's Inequality

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2. Use the Bounded Convergence Theorem to prove Fatou's Lemma
and the Monotone Convergence Theorem for Lebesgue integration.

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3. Prove that there is a subset of $[0,1]$ which is not
Lebesgue measurable. Emphasize where you use the Axiom of Choice.

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4. Evaluate

$$\lim_{n \rightarrow \infty}\int_{(-\infty,\infty)}
\left({510+1488(\sin x)^{12222}\over 1998}\right)^n
{1 \over 1+x^2}\ dx$$

Justify your conclusion.

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5.a. Define what it means for a function $f$ to be of bounded
variation on $[a,b]$.

b. Prove that a function $f$ is of bounded variation on $[a,b]$
if and only if $f$ is the difference of two monotone real valued
functions on $[a,b]$.

c. Find a continuous function on $[0,1]$ which is not a function
of bounded variation, and prove that this function is not of
bounded variation.

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6.a. Give an example of a sequence $f_n$ of {\it continuous} 
functions converging pointwise to a {\it continuous} function
$f$ on $[0,1]$ such that
$$\int_{[0,1]}f_n \not\rightarrow \int_{[0,1]}f$$

b. State the Bounded Convergence Theorem, and explain why the
example in part a does not give a counterexample to this
theorem.

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7.a. Define what it means for an extended real-valued function
$f(x)$ to be (Lebesgue) measurable.  This definition is connected
to four equivalent statements; prove the equivalences.

b. Use the definition in part a to prove or disprove the 
following statement. If $f$ and $g$ are real-valued measurable
functions, then $\{x:f(x)\le g(x)\}$ is measurable.

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