Exercise 11.11

Question

If $f,g \in \mathscr{L}(\mu)$, define the distance between $f$ and $g$ by
    \begin{equation*}
      \int_X \lvert f - g \rvert \, d\mu.
    \end{equation*}
    Prove that $\mathscr{L}(\mu)$ is a complete metric space.

Amit Awasthi · Accepted Answer

\begin{proof}
      Let $\{f_n\}$ be a Cauchy sequence, and $f$ the function constructed from
      a subsequence $\{f_{n_k}\}$ as in the proof of Theorem $11.42$. We want to
      first show $f - f_{n_k} \in \mathscr{L}(\mu)$ for $k$ large enough. Let
      $N$ such that $\int_X \lvert f_n - f_m \rvert < \epsilon$ for all $n,m \ge 
      N$. Then, if $n_k > N$, by Fatou's theorem (Theorem $11.31$),
      \begin{equation*}
        \int_X \lvert f - f_{n_k} \rvert\,d\mu \le \liminf_{i \to \infty}
        \int_X \lvert f_{n_i} - f_{n_k} \rvert\,d\mu < \epsilon.
      \end{equation*}
      Thus, $f - f_{n_k} \in \mathscr{L}(\mu)$ and so $f$ is as well. Moreover,
      since $\epsilon$ was arbitrary, we see
      $$\lim_{k \to \infty} \int_X \lvert f - f_{n_k} \rvert\,d\mu = 0.$$
      We finally know $\{f_n\}$ converges to $f$ in $\mathscr{L}(\mu)$ since
      \begin{equation*}
        \int_X \lvert f - f_n \rvert\,d\mu \le \int_X \lvert f - f_{n_k}
        \rvert\,d\mu + \int_X \lvert f_{n_k} - f_n \rvert\,d\mu,
      \end{equation*}
      and both terms on the right hand side can be made arbitrarily small.
    \end{proof}

Exercise 11.11

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Exercise 11.11

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