Math 6210 • Real Analysis

\begin{enumerate}

\item
State the Monotone Convergence Theorem.


\item
Let \(f : [0, \infty) \to \mathbb{R}\) defined by \(f(x) = e^{-x}\). Explain why \(f\) is
Lebesgue integrable and compute its integral. You may use the
fact that if a function \(f : [0, \infty) \to \mathbb{R}\) is 0 outside an interval \([a, b]\)
and \(f|_{[a,b]} : [a, b] \to \mathbb{R}\) is Riemann integrable then \(f\) is Lebesgue
integrable and the Lebesgue integral of \(f\) equals the Riemann integral of \(f|_{[a,b]}\).


\end{enumerate}