Let $M$ be a Riemannian manifold. I am interested in the dual of the space of continuous (not necessarily smooth) vector field on $M$.
The dual of the space of smooth vector field would be the space of 1-currents because, to my understanding, the space of 1-forms is isomorphic to the space of smooth vector fields using a Riemannian metric. However, the space I am interested in is larger than this because I am not assuming smoothness.
I was wondering there are theorems similar to Riesz-Markov-Kakutani, which talks about the equivalence between Radon measures and the elements of the dual of compactly supported continuous functions.
Is there anything known about this space? Pardon my ignorance if it is covered in Federer or something - I haven't digged too much about the theory of currents.