A character on an infinite group is a positive-definite class function. Extreme characters correspond to traces of finite factor representations of the group in question. Characters can also arise from ergodic mpm actions of groups by taking the measure of fixed points for group elements. In the talk, we will present two results describing the characters for two classes of groups: (1) full groups of Bratteli diagrams and (2) Thompson's groups. The structure of the characters will be related to the structure of ergodic measures for the natural actions of these groups on the spaces they are defined.