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Let $G$ be a Lie group (maybe LCH and second countable topological group is enough here) and $\mu$ be a Borel probability measure on $G$ (not necessarily Haar). Suppose $U$ is an open subset of $G$ of positive measure. I wonder if the follow assertion is true:

There exists open subsets $V,W$ of positive measure such that $VW\subset U$.

My thoughts: certainly I can find open subsets $V,W$ with $VW\subset U$ by the continuity of the multiplication, but I don't see whether $V,W$ have positive measure.

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  • $\begingroup$ I think you can do this using regularity $\endgroup$ – MathQED Sep 23 at 21:23
  • $\begingroup$ @MathQED sorry I don't see how to use the regularity here. Can you say something more? $\endgroup$ – No One Sep 23 at 22:02
  • $\begingroup$ Nevermind my comment. I think it does not work. $\endgroup$ – MathQED Sep 23 at 22:11
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Consider $G$ to be the real line and any measure supported on the interval $U=(100,101)$. If your two sets $V$ and $W$ have positive measure their sum is not contained in $U$.

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  • $\begingroup$ A comment for myself: if the measure is supported on the interval $𝑈=(100,101)$ then any positive subsets $V,W$ must intersect $(100,101)$ and the sum will exceed 200 $\endgroup$ – No One Sep 24 at 14:51
  • $\begingroup$ That is precisely it! $\endgroup$ – Ruy Sep 24 at 14:53

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