The author characterizes removable sets for holomorphic functions in terms of their Hausdorff measures. In particular, he proves the following two general theorems:
1. Let \(\Lambda^{\alpha}\) denote the standard \(\alpha\)-dimensional Hausdorff measure in \({\mathbb{R}}^{2N}\). If E is a closed subset of a k- dimensional complex submanifold \(M\subset {\mathbb{C}}^ N\) such that \(\Lambda^{2k-2}(E)<\infty\) then \({\mathcal O}(M\setminus E)={\mathcal O}(M)\), i.e. E is removable.
2. Let B be the unit ball in \({\mathbb{C}}^ N\) and let \(\Lambda_ B^{\alpha}\) denote the \(\alpha\)-dimensional Hausdorff measure with respect to the distance generated by the Bergman metric in B. If E is a closed subset of B with \(\Lambda_ B^{2N-2}(E)<\infty\) then E is removable.