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\centerline{\bf{Preliminary Examination}}

\centerline{\bf{Real Analysis}}

\centerline{\bf{September 1993}}

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\flushpar Do all of problems 1-2, and as many of the
remaining problems as possible.

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\item{1.} \underbar{State the following theorems}.

(a) Monotone Convergence Theorem

(b) Fatou's Lemma

(c) Dominated Convergence Theorem

(d) H\"older's Inequality

(e) Fubini's Theorem

(f) Egoroff's Theorem

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\item{2.} Prove 1(a), (b), and (c) \underbar{in that order}.

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\item{3.} Give an example of a sequence $f_n$ of
\underbar{continuous} functions on [0,1] converging
pointwise to a \underbar{continuous} function $f$ on [0,1]
such that $\dsize\int f_n \not\to \dsize\int f$.

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\item{4.} Define a sequence of measures $(\mu_n)$ on the
Lebesgue measurable subsets of [0,1] by $\mu_n(A)=\dsize\int^1_0 I_A(x) \ \dsize\frac{1}{n} \ x^{1/n \ - \ 1} dx, \ n=1,2,3,\dots$ .

\item\item{(a)} Verify that $\dsize\lim_{n\to \infty}
\mu_n([a,b])=0$ if $0<a<b<1$.

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\item\item{(b)} Suppose $0<a_k<b_k<1$ and $[a_k,b_k]$ are
disjoint, $k=1,2,\dots,r$.  Verify $\dsize\lim_{n\to \infty} \mu_n(\bigcup^r_{k=1} [a_k,b_k])=0$.

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\item\item{(c)} Suppose $[a_k,b_k]$, $k=1,2,3,\dots$, are all disjoint.  Does $\dsize\lim_{n\to \infty} \mu_n(\bigcup^\infty_{k=1} [a_k,b_k])$ exist?  Give reasoning for your answer.

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\item{5.} Let $m$ be Lebesgue measure on $X = [0,1]$, and let $\mu$ be a measure on the Lebesgue sets with $\mu(X)=1$, and $\mu\sim m$ (i.e. $\mu$ and $m$ have the same sets of measure zero).  Prove there exists a measurable set $A$ such that $\mu(A)=1/2$.

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\item{6.} Let ${\cal{M}}$ be a class of subsets of a set $X$.  ${\cal{M}}$ is a monotone class if $M_n \in {\cal{M}}$ and either $M_n \uparrow M$ or $M_n \downarrow M$ implies $M \in {\cal{M}}$.  Show that if a monotone class ${\cal{M}}$ contains an algebra ${\cal{A}}$, then ${\cal{M}}$ contains the $\sigma$-algebra ${\cal{B}}$ generated by ${\cal{A}}$.

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\item{7.} Construct a sequence of non-negative functions $f_n$ on [0,1] such that $\dsize\int^1_0 f_n(x) dx = 1$ for all $n$, $\dsize\lim_{n\to \infty} f_n(x)=0$ a.e., and for each $g \in C[0,1]$, $\dsize\lim_{n\to \infty} \int^1_0 f_n(x) \ g(x) \ dx = (g(1/5)+g(2/5)+g(3/5))/3.$  (Be sure to verify the last statement.)

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\item{8.} Let $E$ be a subset of [0,1] of Lebesgue measure 0.  Show that there is an absolutely continuous function $f$ whose derivative $f'(x)=+\infty$ for each $x\in E$.

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\item{9.} Let $\{f_n\}^\infty_{n=1}$ and $f$ be non-decreasing functions on [0,1] and suppose the $f_n$ converge in measure to $f$.  Prove that $\lim f_n(x)=f(x)$ at every point, x, where $f$ is continuous.

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\item{10.} Let $f(x)\geq 0$ for each $x\in [0,1]$, $f$ bounded, and suppose that \newline
$E=\{(x,y): 0\leq y \leq f(x)\}$ restricted to $0\leq x \leq 1$ is a measurable subset of the plane.  Prove that $f$ is Lebesgue integrable over [0,1].
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