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\textbf{MATH 501, Analysis, Homework 5}
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\underline{\textbf{1. (3 pts)}} Let $\mathcal{C} \subseteq [0,1]$ be the Cantor set and let $f:[0,1] \rightarrow \mathbb{R}$ be given by
\[ f(x)=\begin{cases} \displaystyle 0 & \text{ if } x \in \mathcal{C}\\ x & \text{ if } x \notin \mathcal{C} \end{cases}\]
Show that $f$ is Riemann integrable and find $\int_0^1 f(x)dx$.\\


\underline{\textbf{2. (2 pts)}} Let $(\X,\M,\mu)$ be a measure space. Let $f,g \in L(\X,\M,\mu)$ and let $(f_n)_{n \in \mathbb{N}}$ and $(g_n)_{n \in \mathbb{N}}$ be sequences of functions in $L(\X,\M,\mu)$. Prove that if $f_n \longrightarrow f$ in measure and $g_n \longrightarrow g$ in measure, then $f_n+g_n \longrightarrow f+g$ in measure.\\

\underline{\textbf{3. (2 pts)}} Consider the measure space $(\mathbb{N},\mathcal{P}(\mathbb{N}),\mu)$ where $\mu$ is the counting measure. Let $f \in L(\mathbb{N},\mathcal{P}(\mathbb{N}),\mu)$ and let $(f_n)_{n \in \mathbb{N}}$ be a sequence of functions in $L(\mathbb{N},\mathcal{P}(\mathbb{N}),\mu)$. Show that if $f_n$'s and $f$ are \textbf{integer valued} and $f_n \rightarrow f$ in $L^1$, then there exists $k \in \mathbb{N}$ such that for all $n \geq k$ we have $f_n=f$.\\

\underline{\textbf{4. (3 pts)}} Let $(\X,\M,\mu)$ be a probability space. Consider the function $$\rho: L(\X,\M,\mu) \times L(\X,\M,\mu) \rightarrow [0,1]$$ given by $$\rho(f,g)=\int_X \min\{|f-g|,1\}\ d\mu$$
Let $f \in L(\X,\M,\mu)$ and let $(f_n)$ be a sequence of functions in $L(\X,\M,\mu)$. Show that if $f_n \rightarrow f$ in measure, then for all $\epsilon \in \mathbb{R}^+$ there exists $k \in \mathbb{N}$ such that for all $n \in \mathbb{N}$ with $n \geq k$, we have that $\rho(f_n,f)<\epsilon$.\\
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\textbf{Solutions}
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