I read multiple times that for a topological space $(\Omega,O)$ with countable basis $F$ and Borel-measure $\mu$, "obviously", the only open null set is the empty set. I don't find that so obvious. Can anyone explain that to me?
2026-02-22 22:33:20.1771799600
Why is $\emptyset$ the only open null set?
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Without further restrictions, it's not true in general. For a silly example, consider the set $X=\{a, b\}$ with the discrete topology, and the Borel measure $\mu$ on $X$ given by $$\mu(\emptyset)=\mu(\{a\})=0,\quad \mu(\{b\})=\mu(X)=1.$$