An intersection functional on the space of subset currents on a free group

Kapovich and Nagnibeda introduced the space (Formula presented.) of subset currents on a free group (Formula presented.) of rank (Formula presented.), which can be thought of as a measure-theoretic completion of the set of all conjugacy classes of finitely generated subgroups of (Formula presented.). We define a product (Formula presented.) of two finitely generated subgroups (Formula presented.) and (Formula presented.) of (Formula presented.) by the sum of the reduced rank (Formula presented.) over all double cosets (Formula presented.), and extend the product (Formula presented.) to a continuous symmetric (Formula presented.)-bilinear functional (Formula presented.). We also give an answer to a question presented by Kapovich and Nagnibeda. The definition of (Formula presented.) originates in the Strengthened Hanna Neumann Conjecture, which has been proven independently by Friedman and Mineyev, and can be stated as follows: for any finitely generated subgroups (Formula presented.) the inequality (Formula presented.) holds. As a corollary to our theorem, this inequality is generalized to the inequality for subset currents.