I've had this exact same experience, the emails are fake but its pretty hard to tell if you aren't savvy. I've also had an anti-scam where someone legit sent me $200 saying it was a mistake and it was for someone else (I have a common seeming email address) and they needed it back because it was for their grandmothers funeral. I said that this sounds like a scam and to contact me in a month to make sure the transfer isn't reversed. I never heard from them again but the $200 stayed in my account.