I’m trying to find out what the chain is that is hanging on one of the exit doors in the interior of the Mayberry courthouse on the anti-Griffith show it’s in all the black-and-white and some of the color episodes. It looks like a long chain, but I don’t know what it’s there for Any help help or ideas would be much appreciated.

It's a device to lock the door into the top of the door frame. I believe it's called a "chain bolt." If you look at the bottom of the door, there's a less complicated device that you can use to lock the door into the bottom of the door frame. Not everyone can reach the top of the door, so they made something clever to lock/unlock it at the top.