If v is a signed measure, E is v-null iff $\displaystyle \mid v \mid (E) = 0 $.

My proof.

If E is v-null, then $\displaystyle v(E)=0$, and $\displaystyle 0=v(E)=v^+(E)-v^-(E)$ by the Jordan Decomposition theorem. Therefore $\displaystyle \mid v \mid (E)= v^+(E)+v^-(E) $ is 0? It just seem like something is missing.

Would anyone please tell me what have I miss here? thank you.