This problem was from Corollary 1 of Lebesgue Nikodym Thorem.
Let $m$ be an usigned $\sigma$-finite measure, and let $\mu$ be a signed $\sigma$-finite measure.
$\mu=m_f$ for some measurable $f$. ($m_f := \int f \, dm $)
For any $\varepsilon > 0$ there exists $\delta >0$ such that $|\mu(E)| < \varepsilon$ whenever $m(E) \le \delta$.
Prof Tao wrote 1 => 2. I looked at ex 11, and did the problem and I needed absolutely integrability of $f$.
I suppose $\sigma$-finiteness of $\mu$ may help, decomposing it to its components still does not work. What am I missing?