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 ∈ M_b(ℝ^d;ℝ^d) such that *div F=0 in the sense of distributions, there exist oriented C^1 loops Γ_i,l with associated measures μ_Γ_i,l such that F= lim_l →∞F_M_b(ℝ^d;ℝ^d)/n_l · l∑_i=1^n_lμ_Γ_i,l weakly-star in the sense of measures and lim_l →∞1/n_l · l∑_i=1^n_lμ_Γ_i,l_M_b(ℝ^d;ℝ^d) = 1. The second, which is an almost immediate consequence of the first, is that smooth compactly supported functions are dense in { F ∈ M_b(ℝ^d;ℝ^d): *divF=0 } with respect to the strict topology.