Gradient lines and distributions of functionals in infinite dimensional Euclidean spaces

This chapter examines the distributions of additive Loeb measures. It shows how to use classical differential geometry to define gradient lines. The chapter shows the connection of these gradient lines to the distribution of a functional. A theorem is proved for the criterion of absolute continuity with respect to Lebesgue measure. This criterion involves the Jacobian of the transformation along gradient lines. However, to make such result more usable, we need to formulate them in more naturally verifiable terms. This is done by producing theorems using geometrical language. More variants of these results, including some for liftings, are also proved.