I have a feeling this should work, but I can't find a proof. All measures are assumed to be regular Borel measures here.
If $\mu = \sum_{i=1}^\infty \mu_i$ is a series of mutually singular measures on some compact Hausdorff space $X$, that converges in the weak*-topology (or vague topology), that is $$\mu(f) = \sum_{i=1}^\infty \mu_i(f) \qquad \text{for all } f\in C(X). $$
Can we say that for a generic $f\in L^2(X, \mu)$ we have the inequality $$||f||^2_\mu \geq \sum_{i=1}^\infty ||f||^2_{\mu_i}$$
where the norms are the $L^2$-norms? Does it hold if the $\mu_i$'s are finitely supported or/and $\mu$ is a probability measure and the space is sufficiently nice (like a compact metric spaces)?
If anyone reads this, the answer is unfortunately no. For metric spaces the Portmanteau theorem gives conditions under which the inequality holds.