The analytic content is not semiadditive

Abstract

We show that the analytic content \(\lambda (\cdot )\) is neither subadditive nor semiadditive. To be precise, for compact sets K in the complex plane, \(\lambda (K)\) is the K-uniform distance from the complex conjugation to the algebra of all rational functions with poles outside K. Thus, given any integer \(n\ge 1\), it is proven that each compactum K can be decomposed as the union of two new compact sets \(E_1\) and \(E_2\) with \(\lambda (E_j)\le 1/n\) for \(j=1,2\). Moreover, we also show that no compactum K with positive analytic content can be decomposed as the countable union of compact sets of zero analytic content.