Please solve the problems: \noindent (1) Let $(\mathbb{R}, \mathcal{A}, \mu)$ be the Lebesgue space (so $\mu$ is the Lebesgue measure and $\mathcal{A}$ are the Lebesgue measurable sets).

\begin{itemize}
    \item[(a)] Let $f_1(x) = \chi_{[0,1]}(x)$; $f_2(x) = \chi_{[0,1/2]}(x)$, $f_3(x) = \chi_{[1/2,1]}(x)$; $f_4(x) = \chi_{[0,1/4]}(x)$,
    $f_5(x) = \chi_{[1/4,2/4]}(x)$, $f_6(x) = \chi_{[2/4,3/4]}(x)$, $f_7(x) = \chi_{[3/4,1]}(x)$; and generally
    \[
    f_n(x) = \chi_{\left[\frac{n - 2^k}{2^k}, \frac{n - 2^k + 1}{2^k}\right]}(x) \quad \text{if } 2^k \leq n < 2^{k+1}.
    \]
    \begin{itemize}
        \item[(i)] Does $(f_n)$ converge to $0$ in measure?
        \item[(ii)] Does $(f_n)$ converge to $0$ in the mean?
        \item[(iii)] Does $(f_n)$ converge to $0$ almost everywhere?
    \end{itemize}
    \item[(b)] Let $g_n(x) = \chi_{[n, n+1]}(x)$ for $n \geq 1$.
    \begin{itemize}
        \item[(i)] Does $(g_n)$ converge to $0$ in measure?
        \item[(ii)] Does $(g_n)$ converge to $0$ almost everywhere?
        \item[(iii)] Does this contradict Corollary 2.4.2? Explain!
    \end{itemize}
\end{itemize}

\noindent  \textbf{Solution:}\\
(a) (i)



\noindent
(2) Let $X \neq \emptyset$ be a set. Fix an element $a \in X$, and define $\mu : \mathcal{P}(X) \to [0, 1]$ by 
\[
\mu(A) = 
\begin{cases} 
1 & \text{if } a \in A, \\
0 & \text{if } a \notin A.
\end{cases}
\]
It is easy to verify that $\mu$ is a measure (you do not have to do that). Show that every function $f : X \to \mathbb{R}$ is integrable and that 
\[
\int f \, d\mu = f(a).
\]
(Here $\mathcal{P}(X)$ denotes the power set of $X$, that is, the set of all subsets of $X$, including $X$ and $\emptyset$.)

\noindent  \textbf{Solution:}\\

\noindent
(3) Let $(\mathbb{R}, \mathcal{A}, \mu)$ be the Lebesgue space. Show that there is no sequence
\[
f_n(x) = a_n \sin(nx) + b_n \cos(nx)
\]
which converges to $1$ almost everywhere on $[0, 2\pi]$ and satisfies $|a_n| + |b_n| \leq 10$ for all $n$.

\noindent  \textbf{Solution:}\\

\noindent
(4) Let $(f_n)$ be a sequence of integrable functions such that $f_n(x) \geq 0$ for all $x$ and for all $n$.
\begin{itemize}
    \item[(a)] Show that 
    \[
    \int \left( \sum_{n=1}^{\infty} f_n \right) d\mu = \sum_{n=1}^{\infty} \int f_n \, d\mu.
    \]
    \item[(b)] Assume that $0 \leq f_{n+1}(x) \leq f_n(x)$ for all $x$ and for all $n$. Show that $\lim_{n \to \infty} f_n = 0$ a.e. if and only if $\lim_{n \to \infty} \int f_n \, d\mu = 0$.
\end{itemize}

\noindent  \textbf{Solution:}\\

\noindent
(5) Let $(X, \mathcal{A}, \mu)$ be a $\sigma$-finite measure space and assume that $f : X \to [0, \infty)$ is an integrable function. Show that for every $\varepsilon > 0$ there exists a set $A \in \mathcal{A}$ such that $\mu(A) < \infty$ and
\[
\varepsilon + \int_A f \, d\mu > \int f \, d\mu.
\]

\noindent  \textbf{Solution:}\\

\noindent
(6) Assume that $\mathcal{A}$ is a $\sigma$-algebra (of subsets of $X$) and that $\mu: \mathcal{A} \to [0, 1]$ and $\nu : \mathcal{A} \to [0, 1]$ are two probability measures (i.e., $\mu(X) = \nu(X) = 1$). Assume further that $\mu(A) = 0$ for every $A \in \mathcal{A}$ for which $\nu(A) = 0$. Let $T : X \to X$ be a transformation such that $T^{-1}(A) \in \mathcal{A}$ for all $A \in \mathcal{A}$. Assume that $T$ preserves both $\mu$ and $\nu$ (i.e., $\mu(T^{-1}(A)) = \mu(A)$ and $\nu(T^{-1}(A)) = \nu(A)$ for all $A \in \mathcal{A}$), and assume that the measure $\nu$ is ergodic\footnote{A measure $\nu$ is ergodic for $T$ if $T^{-1}(A) = A$ implies $\nu(A) = 0$ or $\nu(A) = 1$.} for $T$.
\begin{itemize}
    \item[(a)] Show that $\mu$ is also ergodic for $T$.
    \item[(b)] Show that $\mu = \nu$ (i.e., $\mu(A) = \nu(A)$ for all $A \in \mathcal{A}$). (Hint: Use the Birkhoff Ergodic Theorem with characteristic functions.)
\end{itemize}

\noindent  \textbf{Solution:}\\