Skoda–Zeriahi type integrability and entropy compactness for some measure with L 1 -density

In this article, we prove the Skoda–Zeriahi type integrability theorem with respect to some measure with L 1 -density. In addition, we introduce the log-log threshold in order to detect singularities of Kähler potentials. We prove the positivity of the integrability threshold for such a measure and Kähler potentials with uniform log-log threshold. As an application, we prove the entropy compactness theorem for a family of potential functions of Poincaré type Kähler metrics with uniform log-log threshold. The Ohsawa–Takegoshi L 2 -extension theorem and Skoda–Zeriahi’s integrability theorem play a very important role in this article.