Estimates for integral norms of polynomials on spaces with convex measures

We show that measurable polynomials of degree $d$ are integrable to every positive power and all their $L^p$-norms are equivalent. We also prove a zero-one law for level sets of measurable polynomials and for sets of convergence of measurable polynomials of fixed degree on spaces with convex measures. We obtain an estimate for the $L^1$-norm of continuous polynomials in terms of the $L^1$-norm of their restriction to any set of positive measure.
Bibliography: 19 titles.