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?



Compactness of sigma-algebra for the $L^1$ metrics
Take the classical Bernoulli scheme with the base $(1/2,1/2)$, and let $A\_n$ be the set where the $n$-th coordinate is 1. Then $d(A\_n,A\_m)=1/2$ whenever $n\neq m$. Since all purely non-atomic Lebesgue spaces are isomorphic, the same is applicable to any purely-non-atomic Borel measure on a Polish space.


