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\centerline{\bf{Department of Mathematics and Statistics}}

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\centerline{\bf{Ph.D. Program}}

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\centerline{\bf{Preliminary Examination in Real Analysis}}

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\centerline{\bf{Tuesday, June 8, 1993}}

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\flushpar Directions:  Do all problems except possibly 7, which is an extra credit problem.  The last question is meant to be a longer essay-type question and you should take more time and write more on it.  It is suggested that you do this problem last.

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\item{1.} Let $(X,B,\mu)$ be a measure space, and let
$\{f_n\}$ be a sequence of measurable functions from $X$ into $\bold{R}$, such that $f_n\geq 0$, and $f_n\geq f_{n+1}$, $n=1,2,\dots$  Suppose \newline
$\dsize\lim_{n\to \infty} \int_X f_n(x) \ d\mu(x) =0$.  Show that $\dsize\lim_{n\to \infty} f_n=0$ a.e. on
$X$.

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\item{2.} Define the Cantor ternary set.

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A. Prove that it has power of the continuum.

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B. Prove that it has Lebesgue measure 0.

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C. Prove that its characteristic function is Riemann
integrable on [0,1].

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\item{3.} Let $A_1,A_2,\dots,A_n,\dots$ be a sequence of points of
$R\times R$.  Suppose there is a number $0<r<1$ such that the
distance from $A_n$ to $A_{n+1}$ is $r$ times the distance
from $A_{n-1}$ to $A_n$.  Prove that the sequence of points
is convergent.

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\item{4.} Let $f$ be continuous on the entire real line.
For each positive integer $n$, let $g_n$ be the
characteristic function of $[0, \ 1/n]$ and let $f_n$ be
defined by
$$f_n(x)=n \int^\infty_{-\infty} g_n(x-t) f(t) \ dt \ .$$

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A. Show that $f_n\to f$ pointwise.

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B. Prove or disprove: $f_n$ converges to $f$ uniformly.

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\item{5.} In $R\times R$ let $C$ denote the family of closed
rectangles with sides parallel to the axes and let $D$
denote the family of closed discs (i.e. $E\in D$ if and only
if there is a center $(a,b)$ and a radius $0<r<\infty$ such
that $E=\{(x,y)\}:(x-a)^2+(y-b)^2\leq r\}$.  Prove that the
$\sigma$-ring of sets generated by $C$ coincides with the
$\sigma$-ring generated by $D$.

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\item{6.} In this problem all $L^p$ spaces are formed on the
real line with Lebesgue measure.

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A. If $0<p<q<r\leq \infty$ show that $L^p\cap L^r\subset
L^q$ and that for $f\in L^p\cap L^r$ one has
$$\|f\|_q\leq \|f\|^\lambda_p \|f\|^{1-\lambda}_r$$

\hskip20pt where $\lambda$ satisfies
$\dsize\frac{1}{q}=\dsize\frac{\lambda}{p}+\dsize\frac{1-
\lambda}{r}$.

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B.  Show, by giving appropriate examples, that for
$$0<p<q\leq \infty \ ,$$

\hskip20pt there is no inclusion of either the form $L^p\subset L^q$ or
$L^q\subset L^p$.

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\item{7.}(Extra credit) Let $r_1,r_2,\dots,r_n,\dots$ be an enumeration of
the rational numbers in [0,1].  Let
$$f(x)=\sum_{\{n:x>r_n\}} \frac{1}{2^n} \ .$$
Compute $\dsize\int^1_0 f(x) \ dx$ in terms of $\{r_n\}$.

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\item{8.} A. Write down the axioms for a measure space
$(X,B,\mu)$.

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B. Use your axioms to define the Lebesgue integral,
$$\int_X f(x) \ d\mu(x) \ ,$$

\hskip20pt(for simplicity, it will be enough to do this problem only for 

\hskip20pt non-negative functions.)

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C. Assume $0<\mu(X)<\infty$.  Prove, using the axioms of part A and the definition of 

\hskip18pt part B, the following lemma (from
which most of the convergence theorems follow 

\hskip18pt as an easy consequence).

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Lemma:  Let $\{f_n(x)\}$ be a sequence of integrable
functions.  Suppose there is a constant 

$M$ such that for all $x\in X$ and all $n$,
$$0\leq f_n(x)\leq M<\infty$$

and there is a function $g(x)$ such that $\dsize\lim_{n\to
\infty}f_n(x)=g(x)$ almost everywhere.  

Then $g(x)$ is integrable and
$$\lim_{n\to \infty} \int_X f_n(x) \ d\mu(x) = \int_X g(x) \
d\mu(x) \ .$$

(The idea of the problem is to prove a convergence theorem
from basic principles; in 

particular, Lebesgue dominated convergence or Fatou's lemma should not be used 

unless you have proved them from basic principles as part of this essay.)


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