The analytic content is not semiadditive

We show that the analytic content $$ is neither subadditive nor semiadditive. To be precise, for compact sets K in the complex plane, $$ is the K-uniform distance from the complex conjugation to the algebra of all rational functions with poles outside K. Thus, given any integer $$ , it is proven that each compactum K can be decomposed as the union of two new compact sets $$ and $$ with $$ for $$ . 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.