It is well known that the Haar measure of a Lie group $ G $ arises from a invariant differential form density $ |\omega| $ (of top dimension). Also, we know that if we have a closed subgroup $ H \leq G $, then there is the following criterion:
$ \exists $ a $ G $-invariant (Radon) measure $ \mu $ on $ G/H $ iff $ \Delta_G|_H = \Delta_H $ (modular functions on the groups)
Is there a criterion (in general), when a (arbitrary) measure $ \nu $ on a Lie group $ G $ arises from a smooth differential form or a density? I'm particularly interested in the above case: Does the $ G $-invariant measure $ \mu $ arise from a differential form or a density of the manifold $ G/H $?
I guess there is a positive answer to that question, but I don't see how to show that technically (my differential geometry skills are rather moderate). I think what we somehow should use, is the description of the modular functions $ \Delta_G (g) = \det(Ad(g)) $ in the case of Lie groups.