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We investigate the effect of admitting signed measures as a datum at the scalar Chern-Simons equation $$-\Delta u+{\mathrm{e}}^u({\mathrm{e}}^u-1)=\mu \text{in}\Omega$$ with the Dirichlet boundary condition. Approximating $\mu$ by a sequence ${\left({\mu }_n\right)}_{n\in \mathbb{N}}$ of $L^1$ functions or finite signed measures such that this equation has a solution $u_n$ for each $n\in \mathbb{N}$, we are interested in establishing the convergence of the sequence ${\left(u_n\right)}_{n\in \mathbb{N}}$ to a function $u^{\#}$ and describing the form of the measure which appears on the right-hand side of the scalar Chern-Simons equation solved by $u^{\#}$.