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\title{Problem Set\\ MTH 70200 \\ Real Analysis}
\date{March 14, 2024}

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

\begin{document}
  \maketitle
  \begin{prob}
    Let $f\colon\RR\to\RR$ be a nonnegative Borel-measurable function,
    and let $\varphi(t)=m(\{x\colon f(x)>t\})$, where $m$ denotes Lebesgue measure.
    Show that $\varphi$ is right-continuous and decreasing and that 
    \begin{align*}
      \int_0^{\infty}\varphi(t)\,dt = \int f(x)\,dx.
    \end{align*}
  \end{prob}
  \begin{prob}
    Let $f\in L^1(\RR,\,dm)$, where $m$ denotes Lebesgue measure.
    Adopt the notation $x=(x_1,\ldots,x_n)\in\RR^n$. Let
    \begin{align*}
      A = \bigl\{ x\in\RR^n\colon x_1<\cdots<x_n\bigr\}.
    \end{align*}
    With $m^n$ denoting $n$-dimensional Lebesgue measure, prove that
    \begin{align*}
      \int_A f(x_1)\cdots f(x_n)\,dm^n(x) = \frac{1}{n!}\biggl(\int f(x)\,dx\biggr)^n.
    \end{align*}
    \textit{Hint: consider how the integral behaves under permutation of the coordinates of $x$.}
  \end{prob}
  \begin{prob}
    Let $q_1,q_2,\ldots$ be an enumeration of the rational numbers in $[0,1]$.
    Prove that
    \begin{align*}
      \sum_{n=1}^{\infty}(-1)^n n^{-3/2}\abs{x-q_n}^{\frac12-q_n}
    \end{align*}
    converges to a finite limit for Lebesgue--almost every $x\in[0,1]$.
  \end{prob}
  \begin{prob}(Folland 3.4)
    Suppose that $\nu$ is a signed measure and $\lambda,\mu$ are positive
      measures such that $\nu = \lambda-\mu$. Prove that
      $\lambda\geq\nu^+$ and $\mu\geq\nu^-$.
  \end{prob}
  \begin{prob}(Folland 3.5)
    Suppose that $\nu_1$ and $\nu_2$ are finite signed measures.
    Prove that $|\nu_1+\nu_2|\leq|\nu_1|+|\nu_2|$.
    \textit{Hint: Use the previous problem and you should get a very short solution.}
  \end{prob}
  \begin{prob}
    Show that the distance function $d(\mu,\nu)=\abs{\nu-\mu}(X)$ is a metric
    on the vector space of finite signed measures on $(X,\Mm)$.
  \end{prob}
  
  
  
  
\end{document}
