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

Kapovich and Nagnibeda introduced the space $\mathcal{S} {\rm Curr}(F_N)$ of subset currents on a free group $F_N$ of rank $N\geq 2$, which can be thought of as a measure-theoretic completion of the set of all conjugacy classes of finitely generated subgroups of $F_N$. We define a product $\mathcal{N} (H,K)$ of two finitely generated subgroups $H$ and $K$ of $F_N$ by the sum of the reduced rank $\overline{\rm rk}(H\cap gKg^{-1})$ over all double cosets $HgK\ (g\in F_N)$, and extend the product $\mathcal{N}$ to a continuous symmetric $\mathbb{R}_{\geq 0}$-bilinear functional $\mathcal{N} \colon \mathcal{S} {\rm Curr} (F_N)\times \mathcal{S} {\rm Curr} (F_N)\to \mathbb {R}_{\geq 0}$. We also give an answer to a question presented by Kapovich and Nagnibeda. The definition of $\mathcal{N}$ originates in the Strengthened Hanna Neumann Conjecture, which has been proven by Mineyev and can be stated as follows: $\mathcal{N} (H,K)\leq \overline{{\rm rk}} (H) \overline{\rm rk} (K)$ holds for any finitely generated subgroups $H$ and $K$ of $F_N$. As a corollary to our theorem, this inequality is generalized to the inequality for subset currents.