  The concept of a supercharacter theory of a finite group was introduced by
Diaconis and Isaacs as an alternative to the usual irreducible character
theory, and exemplified with a particular construction in the case of finite
algebra groups. We extend this construction to arbitrary countable discrete
algebra groups, where superclasses and indecomposable supercharacters play the
role of conjugacy classes and indecomposable characters, respectively. Our
construction can be understood as a cruder version of Kirillov's orbit method
and a generalisation of Diaconis and Isaacs construction for finite algebra
groups. However, we adopt an ergodic theoretical point of view. The theory is
then illustrated with the characterisation of the standard supercharacters of
the group of upper unitriangular matrices over an algebraic closed field of
prime characteristic.
