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Let $X$ nonempty set and $\tau$ a compact topology on it. Let $\mu$ be a measure on the Borel $\sigma $- algebra of $\tau$.

Suppose we have a filter $\mathcal{F}$ of clopen sets such that $\mu(f)\geq0.5$ for each $f\in \mathcal{F}$. Is it true that

\begin{equation} \mu \left(\bigcap_{f\in \mathcal{F}} f \right)\geq 0.5 ? \end{equation} or maybe just \begin{equation} \mu\left(\bigcap_{f\in \mathcal{F}} f \right)>0 ? \end{equation}

But if $\tau$ is zero dimensional and compact(a Boolean space) are any of the above true?

For $\mu$ regular and the topology is a Boolean Space, I am able to prove this, but is interesting if it works for $\mu$ any finite measure.

Cezar
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  • Just to be clear, when you define a filter of clopen sets, I assume you mean it is upward closed in the sense of "if $f\subseteq g$ and $f\in \mathcal F$ and $g$ is clopen then $g\in \mathcal F$"? – M W Nov 08 '23 at 00:07
  • Ahh, filter wouldn't be the corect word here. What I mean by filter here is just the family to be closed under finite intersections. So no, without the upwards closure condition. – Cezar Nov 08 '23 at 03:14
  • Gotcha, thanks for clarifying. The counterexample still works, btw. – M W Nov 08 '23 at 05:53

1 Answers1

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Without regularity of $\mu$, the Dieudonné measure on $[0,\omega_1]$ described here gives a counterexample. (At the link it is described on $[0,\omega_1)$ but can be extended to $[0,\omega_1]$ so that $\mu(\{\omega_1\})=0$).

The Dieudonné measure has the property that $\mu(\{\alpha\})=0$ for every $\alpha\in [0,\omega_1]$, yet $\mu(U)=1$ for every neighborhood $U$ of $\omega_1$.

Take $\mathcal F$ to be the family of clopen neighborhoods of $\omega_1$, then $\mu(f)=1$ for each $f\in\mathcal F$, but since every interval of the form $(\alpha,\omega_1]$, $\alpha<\omega_1$ is clopen, we have

$$\mu\left(\bigcap_{f\in \mathcal F}f\right)=\mu\left(\{\omega_1\}\right)=0.$$

M W
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  • Thank you for the example. Do you know of a counterexample when $\mu$ is regular but $\tau$ is not a Boolean Space? – Cezar Nov 08 '23 at 11:09
  • @fsepp If $\tau$ is compact Hausdorff then you should be able to extend your result from the $0$-dimensional case by quotienting out the connected components of your space. The quotient is Boolean, and every clopen set in the original space is saturated, and thus is the complete inverse image of a clopen set in the quotient. Apply the result for Boolean spaces to the push forward measure. You will need regularity of the pushforward measure, which is proved here https://math.stackexchange.com/a/2475119/1210477 – M W Nov 08 '23 at 15:42