***** NOT A TEACHER *****
Hello, Mmhzaf:
I just wanted to remind you that it might be a good idea -- in my OPINION -- not to use that expression until you become more fluent in speaking English. Sometimes "fine by me" can be very rude.
Mona: I'm sorry, but I don't think that we should date anymore. I believe that our interests are so different that we could never have a good relationship. I hope that you understand.
Raul: Fine by me! (= I could not care less! I was thinking about getting a new girlfriend anyway! I'm so glad that I won't be seeing you again!)