Let $F$ be a Frechet space. The main aim of this paper is to prove that $[F^*_{bor}]^*\in (LB_{\infty})$ if and only if $M(X,[F^*_{bor}]^*)=M_{\omega}(X,[F^*_{bor}]^*)$ for every compact uniqueness subset $X$ of $\mathbb C^n$ . We also prove that a compact set $X$ in $\mathbb C^n$ is pluripolar if and only if $X$ is unique and $M(X, F) = M_{\omega}(X, F)$ for every Frechet space $F \in (DN)$.