Abstract: We extend the notion of H-measures on test functions defined on R^d×P, where P⊂R^d is an arbitrary compact simply connected Lipschitz manifold such that there exists a family of regular nonintersecting curves issuing from the manifold and fibrating R^d. We introduce a concept of quasi-solutions to purely fractional scalar conservation laws and apply our extension of the H-measures to prove strong L¹_loc precompactness of such quasi-solutions.