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Let $G$ be a locally compact group with Haar measure $\mu$ (left or right doesn't matter to me). I know that the Haar measure is positive on open sets. What can be said about the Haar measure on singleton sets i.e. sets consisting of a single point? Must it necessarily be zero? For that matter, does the measure of singleton sets even make sense i.e. are singleton sets necessarily Borel sets?

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Singletons are necessarily Borel sets. This is due to the fact that any locally compact group is Hausdorff by definition and hence points are closed in it. Measure of singletons may or may not be zero. The former happens already for the Lebesgue measure on $\mathbb{R}$ and the latter occurs for, say a discrete group, since as you know Haar measure is positive on open sets.

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