Consider the space $\mathcal M$ of all finite complex Borel measures on a segment with norm $\|\mu\|=\int d\,|\mu|$. Assume that a norm-closed linear subspace $\mathcal M_0$ of $\mathcal M$ has the following property:
if $\mu\in\mathcal M_0$ and $f\in L^1(|\mu|)$, then $f\mu\in \mathcal M_0$.
Can $\mathcal M_0$ be characterized as the family of all measures that vanish on a certain fixed collection of subsets of the segment?
Example: the class of all measures $\mu$ such that $d\mu=f\,dx$ for some $f\in L^1$ coincides with the class of measures that vanish on all subsets of zero Lebesgue measure.
Update: For any fixed measure $\mu$, one can define $\mathcal M_0$ as the class of all measures that are absolutely continuous wrt $\mu$. Then the class of subsets in question is the class of subsets of zero $\mu$-measure.