Measure theory (Singularity)

Question

If $μ$ and $v$ be positive measures on $(X,Σ)$ such that each positive $ϵ$ there is a set $A$ in $Σ$ that satisfies $μ(A)<ϵ$ and $ν(A^c)<ϵ$. Now how to prove that $μ$ and $v$ are mutually singular? As far as I understand that I have to choose sets $A_1, A_2, ...$ in such a way that $A=∩ ∪ A_k$, but I can't figure out how to choose $A_k$, please help.

Answer 1

Hint: Start out by taking $A_k$ satisfying $μ(A_k)<2^{-k}$ and $ν(A_k^c)<2^{-k}$, then put $B_n=\bigcup_{k=n}^∞ A_k$. You should find (using countable subadditivity) that $μ(B_n)<2^{1-n}$, while $ν(B_n^c)=0$ (since $B_n^c⊆A_k^c$ for all $k≥n$). Now take $\bigcap_{n=1}^\infty B_n$.