What's an atomic probability space? Can a complete probability space be an atomic probability space?
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An atom of a probability space is a measurable set $A$ with positive measure $P(A)$ and the property that for each measurable subset $B\subseteq A$, either $P(B)=0$ or $P(B)=P(A)$. The probability space is purely atomic if every measurable set with positive measure contains an atom. If $A$ is an atom, it will still be an atom in the completion of the probability space. So yes, a complete probability space can be atomic. Every countable probability space is purely atomic.