Consider a probability space $(X,F,\mu)$, and the quotient $G$ of the sigma-algebra $F$ by its null sets. Endow $G$ with the metrics $d(A,B)=μ(AΔB)$. Is $(G,d)$ a compact metric space?

When $F$ is generated by a (countable) partition, the answer is yes. Is there a general argument to prove that $(G,d)$ is always compact?

Consider a probability space $(X,F,\mu)$, and the quotient $G$ of the sigma-algebra $F$ by its null sets. Endow $G$ with the metric $d(A,B) = \mu(A \triangle B)$. Is $(G,d)$ a compact metric space?

When $F$ is generated by a (countable) partition, the answer is yes. Is there a general argument to prove that $(G,d)$ is always compact?