Positivity criterion for a scalar gradient form (${\int }_{\Omega }c|\nabla v{|}^2\ge 0(v\in C_0^{\infty })⟺c\ge 0\text{ a.e.}$)

Let $c$ be a bounded real measurable coefficient on an open set. Nonnegativity of $c$ clearly makes its gradient form nonnegative. Conversely, if $c<0$ on a set of positive measure, choose an interior Lebesgue point $x_0$ where $c(x_0)<0$. For a nonconstant compactly supported smooth function $\psi$, set $v_{\epsilon }(x)=\epsilon \psi ((x-x_0)/\epsilon )$. Change variables and use the Lebesgue differentiation theorem to obtain ${\epsilon }^{-n}\int c|\nabla v_{\epsilon }{|}^2\to c(x_0)\int |\nabla \psi {|}^2<0$, a contradiction. Thus the linear operator represented by $a(u,v)=\int c\nabla u\cdot \nabla v$ is monotone exactly when $c\ge 0$ almost everywhere.