Given a finite signed measure $\mu$ on a $\sigma$-algebra $\mathcal{A}$, we say that $\mathcal{K}$ is an approximating class for $\mu$ on $\mathcal{A}$ if, for every $A\in\mathcal{A},\varepsilon>0$ there exists $A_\varepsilon\in\mathcal{A}, K_\varepsilon\in\mathcal{K}$ such that $A_\varepsilon\subseteq K_\varepsilon\subseteq A$ and $|\mu(A\setminus A_\varepsilon)|<\varepsilon$.
Now, my question is: if $\mathcal{K}$ is a compact approximating class for $\mu$, is it true that $\mathcal{K}$ is a compact approximating class also for $\mu^+,\mu^-$ (the Jordan decomposition of $\mu$)?
Since every form of triangle inequality doesn't seem useful here, I was thinking about some argument that uses the mutual singularity of $\mu^+$ and $\mu^-$ or other properties of these two measures, because the equality $\mu=\mu^+-\mu^-$ doesn't seem enough.
