Two approximation results for divergence free measures

In this paper, we prove two approximation results for divergence free measures. The first is a form of an assertion of J. Bourgain and H. Brezis concerning the approximation of solenoidal charges in the strict topology: Given F ∈ Mb(Rᵈ;Rᵈ) such that *{div}F=0 in the sense of distributions, there exist oriented C¹ loops Γi,l with associated measures μ_{Γi,l} such that {align*}F= liml → ∞{\|F\|M_b(R^d;R^d)}{nₗ · l}∑ᵢ₌₁n_lμ_{Γi,l}{align*} weakly-star in the sense of measures and {align*}liml → ∞{1}{nₗ · l}∑ᵢ₌₁n_l\|μ_{Γi,l}\|M_b(R^d;R^d)= 1.{align*} The second, which is an almost immediate consequence of the first, is that smooth compactly supported functions are dense in \{ F ∈ Mb(Rᵈ;Rᵈ): *{div}F=0 \} with respect to the strict topology.