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\title{Problem Set\\ MTH 70200 \\ Real Analysis}
\date{February 29, 2024}

\theoremstyle{definition}
\newtheorem{prob}{Problem}

\begin{document}
  \maketitle
  \begin{prob} (Folland 2.14)
    If $f\in L^+(X,\Mm,\mu)$, let $\lambda(E)=\int_E f\,d\mu$ for $E\in\Mm$.
    Show that $\lambda$ is a measure on $\Mm$, and for any $g\in L^+$, 
    we have $\int g\,d\lambda=\int fg\,d\mu$. (Hint: first suppose $g$ is simple.)
  
  \end{prob}
  \begin{prob} (Folland 2.20)
    Suppose that $f_n,g_n,f,g\in L^1$ and $f_n\to f$ a.e.\ and $g_n\to g$\ a.e., and $\abs{f_n}\leq g_n$,
    and $\int g_n\to\int g$. Show that $\int f_n\to\int f$. (Hint: rework the proof of the dominated
    convergence theorem.)
  \end{prob}
  \begin{prob} (Folland 2.21)
    Suppose $f_n,f\in L^1$ and $f_n\to f$ a.e. 
    Show that $\int\abs{f_n-f}\to 0$ iff $\int\abs{f_n}\to\int\abs{f}$. (Hint: use the previous problem.)
  \end{prob}
  
  
  
  \begin{prob}
    Compute
    \begin{align*}
      \lim_{n\to\infty}\int_0^1 n^2x^2e^{-nx}\,dx
    \end{align*}
    and
    \begin{align*}
      \lim_{n\to\infty}\int_0^{\infty}\frac{n\sin\frac{x}{n}}{x(1+x^2)}\,dx.
    \end{align*}
    You can feel free to use elementary calculus facts.
    Please interpret these integrals as Lebesgue integrals with respect to Lebesgue measure.
  \end{prob}
  
\end{document}
2.14
2.20
2.21
2.26
