Given a family of hypersurfaces $H_{t,p} = $ {$x \in \mathbb{R}^n \mid g(x,p) = t $} one defines a generalized Radon transform $R$ of a function $u \colon \mathbb{R}^n \to \mathbb{R}$ as $$ R[u] (t,p) = \int\limits_{H_{t,p}} u(x) a(x) \omega $$ where $\omega$ is such differential form as $d_{x}g \wedge \omega = dx_1 \wedge ... \wedge dx_n$: $$ \omega = \sum\limits_{k=1}^{n} (-1)^{k-1} \frac{\partial_{x_k} g(x,p)}{|\nabla_{x} g(x,p)|^2} dx_1 \wedge ... \wedge \overline{dx_k} \wedge ... \wedge dx_n $$

The first question is why we choose such a form? Which applications provides us this differential form?

The second question concerns the generalisation of this transform to the currents. This transform may be considered as integration of some absolutely continious measure. Is there some generalisation to the case of an arbitrary measure (I think it is related with currents).

Thank You.