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\noindent {\bf Real Analysis Preliminary Exam}  \hfill  {\bf August 26, 2005}
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{\noindent}{\bf Name:} \hspace{60mm} {\bf ID:}\\

1. Do the following things.\\

(a) State the Lebesgue Dominated Convergence Theorem.\\

(b) State Fatou's lemma, and give an example to show that the 
equality may not hold.\\

(c) State the Stone-Weierstrass Theorem.\\

(d) State the Minkowski's Inequality.\\

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2. State and prove the Egoroff's Theorem.

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3. Describe how the Lebesgue measure on ${\bf R}$ is constructed. 

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4. Let $f(x)$ and $g(x)$ be two functions on ${\bf R}$ such that $f$ is continuous and $g$ is measurable. 
Prove that the composition $f(g(x))$ is a measurable function. 

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5. Show that if a monotone sequence $\{f_n\}$ of continuous functions on $[0, 1]$ converges pointwise
to a continuous function, then $\{f_n\}$ converges uniformly. 

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6. Show that the set of irrational numbers is not a countable union of closed subsets of ${\bf R}$. 
(Hint: Use Baire's category theorem.)

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7. Let $f$ be a Lebesgue integrable function over $[0,\ 1]$ and define
$F(x)=\int_{[0,\ x]}fd\lambda$. Show that $F$ is absolutely continuous.

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8. Assume $f\in L^1({\bf R})$ and $E_\epsilon=\{x:\ |f(x)|\geq \epsilon>0\}$.
Show that 
\[\lambda(E_{\epsilon})\leq \frac{\|f\|}{\epsilon}.\]

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9. Assume $0<p<q\leq \infty$. If $f$ is in $L^q([0,\ 1])$, then show that $\|f\|_p\leq \|f\|_q$.

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10. Let $f(x,y)$ be a function on the unit square $D=[0,1]\times [0,1]$ such that for every $x\in [0,1]$ \ $f(x,y)$ is an integrable function of $y$, and 
$$ \int_0^1 f(x,y)dx=g(x) $$ is a continuous function of $x$. 
True, or false: This implies that $f$ is integrable on $D$ and $$ \int_D f(x,y)dxdy=\int_0^1 g(x)dx \ ? $$ 


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