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\textbf{MATH 501, Analysis, Homework 4}
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\underline{\textbf{1. (2 pts)}} Let $(X,\mathcal{M},\mu)$ be a finite measure space and let $f, f_n \in L^1(X,\mathcal{M},\mu)$ be functions such that $f_n \rightarrow f$ uniformly. Prove that $\int_X f_n\ d\mu \rightarrow \int_X f\ d\mu$.\\


\underline{\textbf{2. (2+2 pts)}} Compute the following limits.
\begin{itemize}
\item $\displaystyle \lim_{k \rightarrow \infty}\ \ \int_{[0,1]} \frac{1+kx^2}{(1+x^2)^k}\ d\mathbf{m}$
\item $\displaystyle \lim_{n \rightarrow \infty}\ \ \int_{(0,\infty)} \frac{n\sin(x/n)}{x(1+x/n)^n}\ d\mathbf{m}$\\
\end{itemize}

\underline{\textbf{3. (2+2 pts)}} Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be given by
\[ f(x)=\begin{cases} \displaystyle \frac{1}{\sqrt{x}} & \text{ if } x \in (0,1)\\ 0 & \text{ otherwise}\end{cases}\]
Let $(r_n)_{n \in \mathbb{N}}$ be an enumeration of $\mathbb{Q}$ and consider the measurable function $g: \mathbb{R} \rightarrow \overline{\mathbb{R}}$ given by
\[ g(x)=\sum_{n=0}^{\infty} \frac{f(x-r_n)}{2^n}\]
Show that

\textbf{a.} $g \in L^1(\mathbb{R},\mathfrak{L},\mathbf{m})$

\textbf{b.} The function $g$ is unbounded on every interval.
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\textbf{Solutions}
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