\documentclass{article}

\title{\textbf{Homework 6}}
%\author{Enrique Trevi\~no}

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\begin{document}

\maketitle
\normalsize

\begin{problem}(\textbf{Exercise 4.3.4})\\
Show that if $G$ is any set with finite outer content,
\begin{align*}
G_1\subseteq G_2\subseteq G_3\subseteq \ldots, \text{ and } \bigcup_{k=1}^{\infty} G_k = G,
\end{align*}
then
\begin{align*}
\lim_{k\rightarrow\infty} c_e(G_k)\le c_e(G).
\end{align*}
\end{problem}
\begin{sol}
\end{sol}

\begin{problem}(\textbf{Exercise 4.3.5})\\
Show that if $F$ is any open set,
\begin{align*}
F_1\supseteq F_2\supseteq F_3\supseteq \ldots, \text{ and } \bigcap_{k=1}^{\infty} F_k = F,
\end{align*}
and $F_1$ has finite outer content, then
\begin{align*}
\lim_{k\rightarrow\infty} c_e(F_k) = c_e(F).
\end{align*}
\end{problem}
\begin{sol}
\end{sol}

\begin{problem}(\textbf{Exercise 4.3.8})\\
Show that $x$ is not a $\Gamma$-point if and only if for each $\alpha>0$, there is a neighborhood of $x$, $N_{\varepsilon}(x)$, and a positive integer $N$ such that for all $y\in N_{\varepsilon}(x)$ and all $n\ge N$, $|f_n(y)| < \alpha$.
\end{problem}
\begin{sol}
\end{sol}

\begin{problem}(\textbf{Exercise 4.3.9})\\
Consider the sequence of functions $(f_n)$ defined on $[-1,1]$ by
\begin{equation*}
f_n(x) = \begin{cases} 0, &\text{if } x = 0 \text{ or } |x| > \frac{2}{n},\\
\frac{1}{n}\sin\left(\frac{\pi}{x}\right), & 0 < |x| < \frac{1}{n},\\
\sin\left(\frac{\pi}{x}\right), & \frac{1}{n} \le |x| \le \frac{2}{n}.
\end{cases}
\end{equation*}
Show that this sequence converges to 0 for all $x\in(-1, 1)$. Show that this convergence is not uniform in any $\varepsilon$-neighborhood of 0. For what values of $\alpha > 0$ is 0 in $\Gamma_{\alpha}$?
\end{problem}
\begin{sol}
\end{sol}

\begin{problem}(\textbf{Exercise 4.3.10})\\
Consider the sequence of functions, $(g_n)_{n=1}^{\infty}$ defined on $(-1,1)$ by $g_n(0) = 0$ and for $x\ne 0$,
\begin{equation*}
g_n(x) = \frac{1}{m-1}\left[\left(|x| - \frac{1}{m}\right)m (m-1)\right]^n, \ \ \ \ \ \frac{1}{m} \le |x| < \frac{1}{m-1}, \ \ \ m \ge 2.
\end{equation*}
Show that this sequence converges to 0 for all  $x\in(-1,1)$. Show that this convergence is not uniform in any $\varepsilon$-neighborhood of 0. Show that 0 is an accumulation point of $\Gamma$-points, but it is not a $\Gamma$ point. This demonstrates that while each set $\Gamma_{\alpha}$ is closed, the union of these sets need not be closed.
\end{problem}
\begin{sol}
\end{sol}

\begin{problem}(\textbf{Exercise 4.4.1})\\
Consider the sequence of functions $(f_n)$ defined in Example 4.5 on p. 100. Describe the sets
$$E(n, \sigma) = \{x\in[0,1]\,|\,\, |f_n(x) - f(x)|\ge \sigma\}.$$
Find the value of $\displaystyle\lim_{n\rightarrow \infty} c_e(E(n,\sigma))$.
\end{problem}
\begin{sol}
\end{sol}

\begin{problem}(\textbf{Exercise 4.4.2})\\
Give an example of a sequence of integrable functions on $[0,1]$ that converge to an integrable function and such that
$$\lim_{n\rightarrow\infty} c_e(E(n,\sigma)) = 0,$$
for all $\sigma > 0$, but such that
$$\int_a^b f(x)\,\mathrm{d}x \ne \lim_{n\rightarrow\infty}\int_a^b f_n(x)\,\mathrm{d}x.$$
Thus, we really do need the hypothesis that $|f_n - f|$ is bounded.
\end{problem}
\begin{sol}
\end{sol}

\begin{problem}(\textbf{Exercise 4.4.7})\\
Prove that any subset of a first-category set is of first category.
\end{problem}
\begin{sol}
\end{sol}

\begin{problem}(\textbf{Exercise 4.4.8})\\
Show that if $S$ is of first category and $I$ is an open interval, then $S^{C} \cap I$ cannot be a countable set.
\end{problem}
\begin{sol}
\end{sol}

\begin{problem}(\textbf{Exercise 4.4.11})\\
Prove that any countable union of first-category sets is first category.
\end{problem}
\begin{sol}
\end{sol}




\end{document} 