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\textbf{MATH 501, Analysis, Homework 3}
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\underline{\textbf{1. (2 pts)}} Let $(X,\mathcal{M},\mu)$ be a measure space and let $f_n: X \rightarrow \mathbb{R}$ be measurable for all $n \in \mathbb{N}$. Prove that $\{x \in X: \lim_{n \rightarrow \infty} f_n(x) \text{ exists }\} \in \mathcal{M}$.\\
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\textbf{Hint.} Recall that for a sequence $(a_n)_{n \in \mathbb{N}}$ of real numbers, $\lim_{n \rightarrow \infty} a_n$ exists if and only if $(a_n)_{n \in \mathbb{N}}$ is Cauchy.\\

\underline{\textbf{2. (2 pts)}} Find an index set $I$ and a family $\{f_i: \mathbb{R} \rightarrow \mathbb{R}\ |\ i \in I\}$ of functions such that
\begin{itemize}
\item $f_i$ is Borel measurable for all $i \in I$,
\item The function $f: \mathbb{R} \rightarrow \mathbb{R}$ given by $$f(x)=\sup_{i \in I} f_i(x)\ \ \text{ for all } x \in \mathbb{R}$$
 is \textit{not} Borel measurable.\\
\end{itemize}

\underline{\textbf{3. (3+3 pts)}} Let $(X,\mathcal{M},\mu)$ be a measure space and fix a map $f \in L^+(X,\mathcal{M},\mu)$. Consider the map $\eta: \mathcal{M} \rightarrow [0,+\infty]$ given by
\[ \eta(E)=\int_E f\ d\mu\]

a) Prove that $\eta$ is a measure on the measurable space $(X,\mathcal{M})$.

b) Prove that, for every $g \in L^+(X,\mathcal{M},\eta)$, we have that
\[ \int g\ d \eta= \int fg\ d \mu\]
\textbf{Hint.} Apply the ``usual trick". First, show that this holds for simple functions and try to extend your result via appropriate tools.
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\textbf{Solutions}
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