We study integrability and equivalence of $L^p$-norms of polynomial chaos elements. Relying on known results for Banach space valued polynomials, a simple technique is presented to obtain integrability results for random elements that are not necessarily limits of Banach space valued polynomials. This enables us to prove integrability results for a large class of seminorms of stochastic processes and to answer, partially, a question raised by C. Borell (1979, Séminaire de Probabilités, XIII, 1-3).

Keywords: integrability; chaos processes; seminorms; regulary varying distributions

AMS Subject Classification: 60G17; 60B11; 60B12; 60E15