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\newcommand{\ep}{\varepsilon} \newcommand{\CT}{\mathrm{C}(\bT)} \newcommand{\rC}{\mathrm{C}} \newcommand{\dlim}{\displaystyle\lim\limits} \newcommand{\one}{\boldsymbol{1}} \renewcommand{\phi}{\varphi} \newcommand{\upchi}{{\raise.35ex\hbox{$\chi$}}} \newenvironment{sbmatrix}{\left[\begin{smallmatrix}}{\end{smallmatrix}\right]} \newcommand{\mt}{\emptyset} \newcommand{\lip}{\langle} \newcommand{\rip}{\rangle} \newcommand{\ip}[1]{\lip #1 \rip} \newcommand{\wh}{\widehat} \newcommand{\sot}{\textsc{sot}} \newcommand{\wot}{\textsc{wot}} \newcounter{exerci}[section] \renewcommand{\theexerci}{\arabic{exerci}.} \newenvironment{Ex}{\begin{list}% {\theexerci\hfill}{\usecounter{exerci}\rightmargin=0pt\leftmargin=12pt% \labelwidth=12pt\labelsep=6pt\itemsep=4ex}}{\end{list}\medbreak} \newenvironment{parts}{\vspace{1ex}\begin{enumerate}\renewcommand{\itemsep}{1ex} \renewcommand{\labelenumi}{(\alph{enumi})}}{\end{enumerate}} \newcommand{\hint}{\textbf{Hint:} } \begin{document} \thispagestyle{empty} %%%%%%% 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}