One can even define a product integral for reasonable subsets of an arbitrary Lie group G. Log is interpreted (almost everywhere) as pulling a group element back to the (linear) tangent space T_e at the identity e. An ordinary integral over a portion of R^n = T_e is calculated. Finally the standard Lie group map exp: T_e -> G is applied to the result. Since all compact Lie groups G are isomorphic to some subgroup of GL(n,R) for some n, (I'm not sure about noncompact G) maybe this turns out to be equivalent to the matrix product integral mentioned by mlb. --Dan