Exercise 2.4

Question

\begin{exercise}
    Examine the proof of the Riesz theorem and prove the following two  statements:
    \begin{enumerate}[label=(\alph*)]
      \item If $E_1 \subset V_1$ and $E_2 \subset V_2$, where $V_1$ and $V_2$
        are disjoint open sets, then $\mu(E_1 \cup E_2) = \mu(E_1) + \mu(E_2)$,
        even if $E_1$ and $E_2$ are not in $\mathfrak{M}$.
    \item If $E \in \mathfrak{M}_F$, then $E = N \cup K_1 \cup K_2 \cup \cdots$,
      where $\{K_i\}$ is a disjoint countable collection of compact sets and
      $\mu(N) = 0$.
    \end{enumerate}

Amit Awasthi · Accepted Answer

\begin{proof}[Proof of $(a)$]
      By Step I of the proof, we have the inequality $\mu(E_1 \cup E_2) \le
      \mu(E_1) + \mu(E_2)$, so it suffices to show the converse. Let $W \supset
      E_1 \cup E_2$ be open. Then, $\mu(W) \ge \mu(W \cap V_1) + \mu(W \cap V_2)
      \ge \mu(E_1) + \mu(E_2)$. Taking the infimum over all $W \supset E_1 \cup
      E_2$, we have $\mu(E_1 \cup E_2) \ge \mu(E_1) + \mu(E_2)$.
    \end{proof}
    \begin{proof}[Proof of $(b)$]
      Since $E = E_0 \in \mathfrak{M}_F$, by Step V there is a compact $K_1$ and an
      open $V_1$ such that $K_1 \subset E_0 \subset V_1$ and $\mu(V_1 \setminus
      K_1) < 1$. By Step VI, since $K_1 \in \mathfrak{M}_F$, we have $E_1
      \coloneqq E_0 \setminus K_1 \in \mathfrak{M}_F$. Inductively, we can
      construct from $E_{n-1}$ a compact set $K_n$ and an open set $V_n$ such that
      $\mu(V_n \setminus K_n) < 1/n$, and a new $E_n \coloneqq E_{n-1} \setminus
      K_n \in \mathfrak{M}_F$. By construction, each $K_n$ is disjoint. We now
      claim that $N \coloneqq E \setminus \bigcup_1^\infty K_n$ has measure zero.
      But we have
      \begin{equation*}
        E \setminus \bigcup_{n=1}^\infty K_n = \bigcap_{n=1}^\infty E \setminus
        K_n \subset \bigcap_{n=1}^\infty V_n \setminus K_n,
      \end{equation*}
      whose measure is $< 1/n$ for each $n$, so $\mu(N) = 0$.
    \end{proof}

Exercise 2.4

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Exercise 2.4

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