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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Tame measures on certain compact sets


Author: Hsuan Pei Lee
Journal: Proc. Amer. Math. Soc. 80 (1980), 61-67
MSC: Primary 46J10; Secondary 46E27
MathSciNet review: 574509
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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

$\displaystyle \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}):\vert{z_1}{\vert^2} + \vert{z_2}{\vert^2} = 1$ and $ {z_1} \in T\} $, where T is a compact subset of $ \{ \vert{z_1}\vert < 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.

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DOI: https://doi.org/10.1090/S0002-9939-1980-0574509-1
Keywords: Tame measure, weak-$ ^ \ast $ dense, local algebra
Article copyright: © Copyright 1980 American Mathematical Society