Certain families of measures on coset-spaces, namely inherited, stable, and pseudo-invariant measures, were defined, and shown to exist, in earlier papers, where Jacobians and factor functions, generalizing the idea of Jacobians in theory of functions of several variables, were also denned. In this paper, the existence is established of exact Jacobians and factor functions, which satisfy certain characteristic identities exactly, without an exceptional set of measure zero. A study is made of how properties of a measure are reflected by properties of the Jacobian or the factor function. Necessary and sufficient conditions are found for a function to be an exact Jacobian for some measure.