. Let $X$ be a compact Kähler manifold of complex dimension $n$ and let $\chi$ be a big $(1,1)$-cohomology class admitting a closed positive current with continuous local potentials. Then for any $L\log^{n+\varepsilon}L$-density $v\geq 0$, $\varepsilon>0$ on $X$ such that $\int_{X}v=\int_{X}\chi^{n}$ there exists a unique closed positive current $T\in MA_{\chi}$ such that $T^{n}=v$. Moreover this current possesses continuous local potentials.