Rectifiability of a class of integralgeometric measures and applications

We resolve a long-standing open problem posed by Federer concerning the rectifiability of the integral geometric measure with exponent p >1, thereby settling a question that has persisted since its formulation. While the main theorem is unchanged from previous versions, the exposition and applications have been substantially revised to highlight the result's consequences for Vitushkin's conjecture on analytic capacity and removability in the complex plane. As an application, we establish two novel results related to Vitushkin's conjecture: in a multi-scale setting, we provide an affirmative answer for sets with finite integral geometric measure within regimes of Favard length behavior at small scales not previously addressed; and in a single-scale framework, we extend the Besicovitch-Federer projection theorem beyond the classical sigma-finite setting, namely for planar sets intersecting a typical line in finitely many points. The rectifiability criterion for general Radon measures via slicing, included in earlier versions, has been removed and will appear in separate work.