I have a question regarding two equal events with probability 1. On page 21 of Papoulis' "Probability, Random Variables, and Stochastic Processes", 4th edition, there is an equation:
$ P(A\bar{B}) + P(\bar{A}B) = 0 \iff P(A) = P(B) = P(AB) $
I am able to prove the relation in the reverse direction, but how do I prove it in the forward direction?
$ P(A\bar{B}) + P(\bar{A}B) = 0 \implies P(A) = P(B) = P(AB) $
My work:
$ P(A\bar{B}) + P(\bar{A}B) = 0 \implies P(A\bar{B}) + P(\bar{A}B) + P(AB) = P(AB)$
$ \implies P(A\bar{B}) + P(B) = P(AB) \implies P(A) + P(B) = 2P(AB)$
For the last step, I added $P(AB)$ to both sides. I am stuck at this point. Can I have some hints on how to proceed? Thank you in advance for your help.