Exercise 1.11

Question

Show that
    \begin{equation*}
      A = \bigcap_{n=1}^\infty \bigcup_{k=n}^\infty E_k
    \end{equation*}
    in Theorem $1.41$, and hence prove the theorem without any reference to
    integration.

Amit Awasthi · Accepted Answer

\begin{proof}
      $x \in A$ if and only if $x$ is in infinitely many $E_n$, which is
      equivalent to saying that for all $n$, $x$ is in some $E_k$ for $k \ge n$.
      Converting this to set notation we get the claim equality of sets.
      \par Now we want to show $\mu(A) = 0$. But we have
      \begin{align*}
        \mu(A) &= \mu\!\left( \bigcap_{n=1}^\infty \bigcup_{k=n}^\infty E_k
        ight)\
        &\le \inf_{n \ge 1} \mu\!\left( \bigcup_{k=n}^\infty E_k ight)\
      &\le \int_{n \ge 1} \sum_{k=n}^\infty \mu(E_k) = 0.\qedhere
      \end{align*}
    \end{proof}

Exercise 1.11

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Exercise 1.11

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