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{\bf MATH 603: Basic Real Analysis I -- FALL 2002}\\

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\begin{Large}

\centerline{\bf FINAL EXAM}

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\begin{tabular}{|c|c|c|}

\hline \hline Problem & Possible points & SCORE \\ \hline \hline

1 & 10 & \\ \hline 2 & 10 & \\ \hline 3 & 10 & \\ \hline 4 & 10 &

\\ \hline 5 & 10 & \\

\hline 6 & 10 & \\ \hline 7 & 10 & \\

\hline \hline

TOTAL & 75 & \\

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\noindent {\bf 1.} Let $f: X \to \mathbb{R}$ be a measurable function.

Prove that for any $p>0$

$$

{\displaystyle

\int_X |f(x)|^p d\mu(x) =\int_0^\infty p t^{p-1} F(t) dt ,

}

$$

where $F(t)=\mu(\{ x: |f(x)| > t\})$.

\medskip



\noindent {\bf 2.} Let $\mu$ be a finite measure on $X$.

Prove that a non-negative function $f$ is in $L^1(X, d\mu)$

if and only if the series

$$

{\displaystyle

\sum_{n=0}^\infty 2^{-n} \mu(\{ x: f(x) >  2^{-n}\})\

}

$$

converges.

\medskip



\noindent {\bf 3.} Give an example of a sequence of functions,

$(f_n)$, Lebesgue integrable on $[0,1]$ and such that\\

(i) $f_n \to f$ a.e.\\

(ii) $\lim_{n\to \infty} \int_{[a,b]} f_n (x) dx =  \int_{[a,b]} f (x)dx$\\

(iii) there is no $g \in L^1([0,1])$ such that, for  $| f_n (x) | \le g(x)$

a.e. $\forall n$.



\medskip



\noindent {\bf 4.} Let $(X,M,\mu)$ be a $\sigma$-finite measure space, $N$ a

sub-$\sigma$-algebra of $M$ and $\nu=\mu\upharpoonright N$.

Prove:\\ If $f \in  L^1(\mu)$, there exists $g \in  L^1(\nu)$ such that

$$

\int_{E} f d\mu =  \int_{E} g d\nu

$$

for all $E\in N$.

\medskip



\noindent {\bf 5.} Two (signed) measures $\nu_1$ and $\nu_2$ are called

{\it equivalent} if $\nu_1 \ll \nu_2$ and $\nu_2 \ll \nu_1$.

Let $\mu$ be a measure on $X$ and $f_1, f_2$ two $\mu$-integrable

real functions on $X$. Define signed measures $d\nu_i= f_i d \mu,\ i=1,2$.

Prove that  $\nu_1$ and $\nu_2$ are equivalent

if and only if $\mu(N_1 \Delta N_2) =0$, where

$N_i=\{ x\in X; f_i(x) \ne 0 \}$.



\noindent {\bf 6.} Let $f$ and $g$ be two functions absolutely

continuous on  $[a,b]$. Prove that the integration by parts formula

holds for $f$ and $g$ :

$${\displaystyle \int_a^b f(x) g'(x) dx = f(x) g(x)\mid_a^b -  \int_a^b f'(x)

g(x) dx\ .}$$











\noindent {\bf 7.} Given two compact metric spaces, $X$ and $Y$, let

$C(X,Y)$

be the set of all continuous mappings of $X$ into $Y$. Define

distance in $C(X,Y)$ by

$$ \rho(f,g) = sup_{x\in X} \rho (f(x), g(x))\ .$$



\medskip

\noi a) Prove that $C(X,Y)$ is a metric space and that it is complete.



\medskip

\noi b) Prove the following generalization of Arzela's theorem:

A set $D\subset C(X,Y)$ is pre-compact if and only if $D$ is an

{\it equicontinuous} family of functions, i. e. , given any $\epsilon >

0$,

there exists a number $\delta > 0$ such that $\rho(x', x'') < \delta $

implies $\rho(f(x'), f(x'')) < \epsilon $ for all $x', x'' \in X$ and

all

$f\in D$.



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Two (signed) measures $\nu_1$ and $\nu_2$ are called

{\it equivalent} if $\nu_1 \ll \nu_2$ and $\nu_2 \ll \nu_1$.

Let $\mu$ be a measure on $X$ and $f_1, f_2$ two $\mu$-integrable

real functions on $X$. Define signed measures $d\nu_i= f_i d \mu,\ i=1,2$.

Prove that  $\nu_1$ and $\nu_2$ are equivalent

if and only if $\mu(N_1 \Delta N_2) =0$, where

$N_i=\{ x\in X; f_i(x) \ne 0 \}$.

