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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.

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.

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.

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R W
R W
  • 17k
  • 3
  • 37
  • 74

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.