Exercise 2.11

Question

Let $\mu$ be a regular Borel measure on a compact Hausdorff space $X$;
    assume $\mu(X) = 1$. Prove that there is a compact set $K \subset X$ (the
    \emph{carrier} or \emph{support} of $\mu$) such that $\mu(K) = 1$ but
    $\mu(H) < 1$ for every proper compact subset $H$ of $K$.

Amit Awasthi · Accepted Answer

\begin{proof}
      %Let $K$ be the intersection of all compact $K_\alpha$ with $\mu(K_\alpha) =
      %1$; then every open set $V$ which contains $K$ also contains some
      %$K_\alpha$. Regularity of $\mu$ is needed. Show that $K^c$ is the largest
      %open set in $X$ whose measure is $0$.
      Let $K$ be the intersection of all compact $K_\alpha$ with
      $\mu(K_\alpha) = 1$. We claim every open set $V$ containing $K$ contains
      some $K_\alpha$, which implies $K$ has measure $1$.
      For, if $V \supset K$,
      then $V^c$ is covered by $\bigcup K_\alpha^c$, and since $V^c$ is closed
      hence compact, it is covered by a finite union $\bigcup_1^N K_n^c$. But
      $\bigcup_1^N K_n^c$ has measure $0$ since it is the finite union of sets
      of measure $0$. Thus, $V$ contains $\bigcap_1^N K_n$, which is of measure
      $1$, hence is of the form $K_\alpha$. Now suppose $H \subsetneq K$, and
      suppose $\mu(H) = 1$. Then, $K \subset H$ by construction, which is a
      contradiction.
    \end{proof}

Exercise 2.11

Answers

Comments

Exercise 2.11

Answers

Comments

Add answer