Suppose $X$ is a locally compact Hausdorff space. Define the Baire sets in $X$, denoted by $\mathcal Ba(X)$, to be the smallest $\sigma$-algebra that contains all compact $G_\delta$ subsets of $X$. Let $x \in X$. Define the Dirac measure $\delta_x$ on $\mathcal Ba(X)$ by $\delta_x(E)$ = 1 if $x \in E$ and 0 otherwise. Must Dirac measure be regular on the Baire sets?
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2$\begingroup$ Cross posted from MSE: math.stackexchange.com/questions/1103444/… $\endgroup$– Asaf Karagila ♦Commented Jan 17, 2015 at 18:00
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1$\begingroup$ $\{x\}$ may not be a $G_\delta$. $\endgroup$– Richard HevenerCommented Jan 17, 2015 at 19:54
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$\begingroup$ So the definition of "regular" here should be: for each Baire set $A \subset X$ and each $\epsilon > 0$, there exists an open Baire set $U$ and a compact Baire set $K$ with $K \subset A \subset U$ and $\mu(U \setminus K) < \epsilon$. And the problem with @ChristianRemling's suggestion is that $\{x\}$, though compact, may not be Baire. Is that right? $\endgroup$– Nate EldredgeCommented Jan 17, 2015 at 21:25
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$\begingroup$ In other words, to address @ChristianRemling's comment, if $\{x\}$ is not Baire (which can happen as in $[0,1]^{[0,1]}$ then $\delta(\{x\})$ and $\delta(X\setminus\{x\})$ are undefined. $\endgroup$– Nate EldredgeCommented Jan 17, 2015 at 21:42
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3$\begingroup$ This question should be closed because it has received a satisfactory answer on math.SE $\endgroup$– Yemon ChoiCommented Jan 18, 2015 at 16:10
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