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 \centerline{\smc Real Analysis Exam, August 1998 }
 \par\medskip \centerline {Part I}
 \par\medskip \centerline {\it Do three problems in this section}
 \par\bigskip\noindent I.1. For $x\in \R$ let $f_n(x) =
\frac{1+x^2}{1+nx^2}$. Show:
 \roster
 \item $f_n$ converges pointwise but not uniformly.
 \item $f_n\to 0$ a.e.
 \item $f_n \to 0$ in measure.
 \endroster
 
 \par\medskip\noindent I.2. Find all values of $p\in \R$ for which
the function $f(x)=|\log x|^p$ belongs to $L^1(0,1)$ with respect to
Lebesgue measure. (Hint when $p\geq 1$: first consider integral $p$
and use integration by parts).

 \par\medskip\noindent I.3. Suppose $f_n:[0,1]\to \R$, $n=0$,
$1,\dots$ is a sequence of continuous functions such that for each
$x \in [0,1]$,
 \roster
 \item  $f_n(x) \to 0$
 \item $f_n(x) \geq f_{n+1}(x)$
 \endroster
 Show that $f_n \to 0$ uniformly.

 \par\medskip\noindent I.4. Let $\Omega=\R\times[-1,1]$, let
$f:\Omega\to \R$ be defined by $f(x,y) = e^{-|x|/y}/y$
for $y \ne 0$. Compute $\int_\Omega f\, dxdy.$
 
 \medskip
% \newpage
 \centerline {Part II}
 \par\medskip \centerline {\it Do two problems in this section}
 \bigskip\noindent II.1. Let $T:\R^2\to\R^2$ be defined by
$T(x,y) = (2x,y/4)$. Show that if $f\in L^1(\R^2)$, and $f\circ
T=f$ then $f = 0$ a.e. 
 
 \par\medskip\noindent II.2. Let $I = [0,1] \subset \R$. The
Weierstrass approximation theorem states that if $f \in C(I)$ then
there is a sequence $\{p_n\}_{n=0}^\infty$ of polynomials such that
$p_n\to f$ uniformly for $x$ in $I$ as $n\to \infty$. Show that if
$f\in C^1(I)$ then there is a sequence $\{p_n\}_{n=0}^\infty$ of
polynomials converging to $f$ in $C^1(I)$, that is, both $p_n\to f$
and $p'_n\to f'$ uniformly for $x$ in $I$ as $n\to \infty$.
 
 \par\medskip\noindent II.3. Let $\Cal B$ be the Borel
$\sigma$-algebra of $\R$, let $\mu=\lambda + \delta$ where $\lambda$
is the Lebesgue measure and $\delta$ is the Dirac measure at $0$,
$\delta(E) = 1$ if $0 \in E$, $\delta(E) = 0$ if $0 \notin E$, for
any $E \in \Cal B$.
 \roster
 \item Show that if $f \in L^\infty(\R,\mu)$ then $f\in 
L^\infty(\R,\lambda)$, and 
 $$\|f\|_{L^\infty(\R,\lambda)} \leq
\|f\|_{L^\infty(\R,\mu)}.$$
 \endroster
 Thus there is a well defined linear mapping 
$T:L^\infty(\R,\mu) \to L^\infty(\R,\lambda)$, simply taking an
element $f$ in $L^\infty(\R,\mu)$ and regarding it as an element of
$L^\infty(\R,\lambda)$, and the mapping is continuous.
 \roster \item "(2)" Show that $T$ is surjective but not
injective. 
 \endroster
 Finally,
 \roster \item "(3)" Show that elements of $L^\infty(\R,\mu)$ have
a well defined value at $0$.
 \endroster
 Hint for \therosteritem2 and \therosteritem3: Keep in mind that
strictly speaking, the elements of $L^\infty(\R,\mu)$ are
equivalence classes of functions.
 \enddocument