Aren't psychiatric wards full of people who believe they are someone else?
As long as the person is able to function, and not distressed and can control their actions then they can think whatever without it being psychiatric.
Cops will try to capture people with this condition. Sometimes illegally.
If they do it illegally, then they are not good law enforces:P
You would be surprised what Cops can get away with.
Ah yes, that is true. They should investigate themselves, they are very good prosecutors of themselves like most other people:P