Let $\nu$ be a non-zero finite (not necessarily positive) measure on a compact Kähler surface $M$.
Is it always possible to decompose the measure $\nu$ as follows $\nu=\lambda_1-\lambda_2$?
Where $\lambda_1$ and $\lambda_2$ are (non-zero) finite positive measures that, in addition, are mutually singular.
You can take a look at the screenshot taken from "Harmonic Currents of Finite Energy and Laminations", J. E. Fornaess and N. Sibony, Page $980$, Proof of Lemma $3.1$.
(We decompose the measure $c(\alpha) \nu(\alpha)$ on the space of plaques, $c(\alpha) \nu(\alpha)=\lambda_1 - \lambda_2$ for positive mutually singular measures $\lambda_j$).
