Tame measures on certain compact sets

Abstract:

A finite complex Borel measure $\mu$ on a compact subset $X \subset {{\mathbf {C}}^n}$ is called tame if there exist finite measures ${\sigma _1}, \ldots ,{\sigma _n}$ on X with \[ \int _X \phi d\mu = \int _X {\sum \limits _1^n {\frac {{\partial \phi }}{{\partial {{\bar z}_j}}}d{\sigma _j}} } \] for every $\phi \in C_0^\infty ({{\mathbf {C}}^n})$. We define ${X_T} = \{ ({z_1},{z_2}):|{z_1}{|^2} + |{z_2}{|^2} = 1$ and ${z_1} \in T\}$, where T is a compact subset of $\{ |{z_1}| < 1\}$ in ${{\mathbf {C}}^1}$. It is shown in this paper that tame measures form a weak-$^ \ast$ dense subset of $R{({X_T})^ \bot }$. It follows then, with the help of a theorem by Weinstock, that $R({X_T})$ is a local algebra.