Abstract. It is known that the Fejér means -- with respect to Vilenkin systems -- of an integrable function on bounded Vilenkin groups converge to the function a.e. In this work we prove the above on similar, not necessarily Abelian, totally disconnected groups with respect to the product system of normed coordinate functions of continuous irreducible unitary representations of the coordinate groups.

AMS Subject Classification (1991): 43A75, 43A55, 22D10

Keyword(s): compact totally disconnected groups, Fejér means, maximal function, pointwise convergence, (H, typeness, L)

Received September 26, 1994, and in revised form April 12, 1995. (Registered under 5634/2009.)