- #1
Euge
Gold Member
MHB
POTW Director
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Here is this week's POTW:
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Let $X$ be a locally compact Hausdorff space, and let $\mu$ be a Radon measure on $X$. Recall that the complement of the support of $\mu$ is the union of all open subsets of $X$ of $\mu$-measure zero. Show that the support of $\mu$ is the set of all $x\in X$ such that for all compactly supported continuous functions $f : X\to [0,1]$ with $f(x) > 0$, the integral $\int_X f\, d\mu > 0$.-----
Remember to read the https://mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to https://mathhelpboards.com/forms.php?do=form&fid=2!
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Let $X$ be a locally compact Hausdorff space, and let $\mu$ be a Radon measure on $X$. Recall that the complement of the support of $\mu$ is the union of all open subsets of $X$ of $\mu$-measure zero. Show that the support of $\mu$ is the set of all $x\in X$ such that for all compactly supported continuous functions $f : X\to [0,1]$ with $f(x) > 0$, the integral $\int_X f\, d\mu > 0$.-----
Remember to read the https://mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to https://mathhelpboards.com/forms.php?do=form&fid=2!