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

My proof.

If E is v-null, then v(E)=0, and 0=v(E)=v^+(E)-v^-(E) by the Jordan Decomposition theorem. Therefore \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.