Iterated path integrals and generalized paths

Let 9J£ be a C°° manifold with a countable basis. For convenience, it is assumed that 2JÎ is Riemannian. Let $ be the set of "reduced" piecewise C paths having a common initial point p in 5DÎ such that each ceG'ip is parameterized by arc length. By a reduced path a: [0, /]—»2)î, we mean one such that there exists no / £ ( 0 , /) with a(t — s) =a(t+s) for \s\ sufficiently small. Let 12 be the vector space (over the real number field R) of C 1-forms on 5DÎ. Elements of S will be denoted by w, wi, w2, • • • . Let a be the restriction a\ [0, t], O^t^lLet fawi be the usual integral, and define, for r>l,