Exercise 2.1

Question

Let $\{f_n\}$ be a sequence of real nonnegative functions on $R^1$, and consider the following four statements:
    \begin{enumerate}[label=(\alph*)]
      \item If $f_1$ and $f_2$ are upper semicontinuous, then $f_1 + f_2$ is
        upper semicontinuous.
      \item If $f_1$ and $f_2$ are lower semicontinuous, then $f_1 + f_2$ is
        lower semicontinuous.
      \item If each $f_n$ is upper semicontinuous, then $\sum_1^\infty f_n$ is
        upper semicontinuous.
      \item If each $f_n$ is lower semicontinuous, then $\sum_1^\infty f_n$ is
        lower semicontinuous.
    \end{enumerate}
    Show that three of these are true and that one is false. What happens if the word ``nonnegative'' is omitted? Is the truth of the statements affected if $R^1$ is replaced by a general topological space?

Amit Awasthi · Accepted Answer

\begin{proof}
      $(a)$ is true without the nonnegative hypothesis nor assuming $f_i$ are
      functions on $\mathbf{R}^1$ since
      \begin{multline*}
        (f_1+f_2)^{-1}([-\infty,\alpha)) = \{x \mid f_1(x) + f_2(x) < \alpha\}\
        = \bigcup_{N \in \mathbf{R}} f_1^{-1}([-\infty,N)) \cap
        f_2^{-1}([-\infty,N-\alpha))
      \end{multline*}
      is open. Likewise, $(b)$ is true without the nonnegative hypothesis nor
      assuming $f_i$ are functions on $\mathbf{R}^1$ since
      \begin{multline*}
        (f_1+f_2)^{-1}((\alpha,\infty]) = \{x \mid f_1(x) + f_2(x) > \alpha\}\
        = \bigcup_{N \in \mathbf{R}} f_1^{-1}((N,\infty]) \cap
        f_2^{-1}((N-\alpha,\infty])
      \end{multline*}
      is open.
      \par Now we claim $(c)$ is false even in the case stated. Let $h_n$ be the
      function defined as
      \begin{equation*}
        h_n(x) = \begin{dcases*}
          0 & if $x \le 0$,\
          nx & if $0 \le x \le 1/n$,\
          1 & if $x \ge 1/n$.
        \end{dcases*}
      \end{equation*}
      Each $h_n$ is continuous by definition. Now let $f_n = h_{n} -
      h_{n-1}$ where $h_0 = 0$. Then, $\sum_{1}^N f_n = h_N$, and so
      \begin{equation*}
        \sum_{n=1}^\infty f_n = \begin{dcases*}
          0 & if $x \le 0$,\
          1 & if $x > 0$
        \end{dcases*}
      \end{equation*}
      is not upper semicontinuous.
      \par Now we claim $(d)$ is true even if $\mathbf{R}^1$ is replaced by a
      general topological space, but not if we remove the condition that the
      $f_n$ are nonnegative. $(d)$ is true under the stated hypotheses since
      $\sum_1^\infty f_n$ is the infimum of the collection of partial sums
      $\sum_1^N f_n$, and then by $(2.8(c))$. On the other hand, we can take the
      negative of the example for $(c)$ to show the hypothesis that the $f_n$
      are nonnegative is necessary.
    \end{proof}

Exercise 2.1

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Exercise 2.1

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