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\flushleft\underline{Homework 12 --- MATH 4590 Spring 2018} \\
Let $m$ denote Lebesgue measure.
\begin{enumerate}[1.]
\item Calculate $m\left( [3,10] \right)$.
\item Calculate $m \left( [1,3] \cup [2,4] \cup [3,5] \right)$.
\item Let $E_1=[1,3], E_2=[2,4], E_3=[3,5]$. Draw a picture of the simple function $\psi \colon \mathbb{R} \rightarrow \mathbb{R}$ defined by
$$\psi(x) = \displaystyle\sum_{k=1}^3 (2k-3)\chi_{E_k}(x).$$
\item Compute the integral of $\psi$ as defined in Problem 3. 
\begin{enumerate}
\item Compute it as a Riemann/Darboux integral.
\item Compute it as a Lebesgue integral (this answer should match the other one, but the definition is different!).
\end{enumerate}
\item Consider the function $\sin \colon \mathbb{R} \rightarrow [-1,1]$. Draw a picture of $\sin^+$ and draw a picture of $\sin^{-}$. 
\end{enumerate}
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