  We investigate the scalar Chern-Simons equation $-\Delta u + e^u(e^u-1) =
\mu$ in cases where there is no solution for a given nonnegative finite measure
$\mu$. Approximating $\mu$ by a sequence of nonnegative $L^1$ functions or
finite measures for which this equation has a solution, we show that the
sequence of solutions of the Dirichlet problem converges to the solution with
largest possible datum $\mu^# \le \mu$ and we derive an explicit formula of
$\mu^#$ in terms of $\mu$. The counterpart for the Chern-Simons system with
datum $(\mu, \nu)$ behaves differently and the conclusion depends on how much
the measures $\mu$ and $\nu$ charge singletons.
