% LaTeX file -- PM 450 Winter 2017, Assignment 5

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\begin{document}
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%%%%%%%   TITLE   %%%%%%
{\noindent\textbf{\large PM 450 \hfill Assignment 5 \hfill Due Wednesday March 15.}} 
\vskip 2em 

\begin{Ex}

\item \begin{parts}
\item Prove that if $f_n\ge 0$ are measurable functions on $X$, then $\dint\liminf f_n \le \liminf \dint f_n$.
\item Suppose that $f_n\ge 0$ are measurable functions on $X$ and $f_n \to f \ae$\ Moreover suppose that
$ \dlim_{n\to\infty} \dint f_n = \dint f < \infty$. Prove that for every measurable $E\subset X$,
$\dlim_{n\to\infty} \dint_E f_n = \dint_E f$. 
\item Construct an example to show that the previous result can fail if $\dint f = \infty$.
\end{parts}
 

\item 
Evaluate the following limits and justify your argument with appropriate convergence theorems.
\begin{parts}
\item $\displaystyle \lim_{n\to\infty} \int_0^\infty \frac{1+nx^2}{(1+x^2)^n} \,dx$
\hspace{30mm} (b)\ $\displaystyle \lim_{n\to\infty} \int_0^\infty \frac{n\sin(x/n)}{x(1+x^2)} \,dx$
\end{parts}


\item Suppose that $f_n \!\to\! f \ae$ and $g_n \!\to\!  g \ae$ are all integrable functions and $|f_n|\le g_n$ for $n\ge1$.
Suppose $\dlim_{n\to\infty} \dint g_n = \dint g$. Prove that $\dlim_{n\to\infty} \dint f_n = \dint f$.\qquad 
\hint rework  the proof of LDCT.


\item 
\begin{parts}
\item Show that the improper Riemann-integrable function $f(x) = \dfrac{\sin x}x$ for $x\ge0$ is not Lebesgue integrable on $[0,\infty)$.
\item Show that if $f$ is a bounded Lebesgue integrable function on $[0,\infty)$ and is also improper Riemann integrable,
then the two integrals agree.
\end{parts}


\item Suppose that $f,\, f_n \in L^p$ for some $1 \le p < \infty$ and $f_n\to f \ae$ and $\|f_n\|_p \to \|f\|_p$.
Prove that $f_n$ converges to $f$ in the $L^p$ norm. 


\item Let $1 \le p < r \le \infty$.
\begin{parts}
\item Suppose that $X \subset \bR$ is measurable, and $m(X)<\infty$. Prove that $L^r(X) \subset L^p(X)$. \\
\hint for $f$ measurable on $X$, show that $\|f \|_p \le \|f \|_r m(X)^{\frac1p-\frac1r}$.
\item Find a function $f$ so that $f\in L^q((0,\infty))$ if and only if $p < q \le r$. \hint consider combinations of functions of the form
$f(x) = x^{-a} |\log x|^{-b}$ on various domains.
\end{parts}


\item Let $A_n = \big\{ x\in [0,1] : x = (0.x_1x_2 \dots)_{\text{base}\,2} \AND x_n=0 \big\}$ for $n\ge 1$.\\
Prove that
$\dlim_{n\to\infty} \dint_{A_n} f = \frac12 \int_{[0,1]} f$
for all integrable functions on $[0,1]$.\\ 
\hint first prove it for step functions with jumps at diadic rationals.



\end{Ex}
\end{document}