If you execute an innocent person, is that murder?

I know the State of Illinois has banned the death penalty because the state legislature ascertained many death row inmates were in fact innocent due to extensive DNA testing. I personally, have mixed feelings about the death penalty. If someone d and murdered my mom, I would want that person to die. First and foremost, I would want the authorities to be absolutely positive they have the right man. So if the judicial system executes the wrong person, is that murder? You can't tell me that hasn't happened before, either. I remember that movie Thin Blue Line about Randall Ward who was proven innocent 12 days before his execution after he exhausted all his appeals. It later turned out the man who actually accused him of the murder was the one who did it. So would you rather a guilty man be free and then retried or would you rather an innocent man be put to death?