# Subsets of positive measure in a group

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.

• I think you can do this using regularity – MathQED Sep 23 at 21:23
• @MathQED sorry I don't see how to use the regularity here. Can you say something more? – No One Sep 23 at 22:02
• Nevermind my comment. I think it does not work. – MathQED Sep 23 at 22:11

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 $$U$$.
• 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 – No One Sep 24 at 14:51