Let A be a uniform algebra on a compact Hausdorff space X and m a probability measure on X. Let HP(m) be the norm closure of A in LP(m) with 1 less than or equal to p less than or equal to infinity and H-infinity(m) the weak * closure of A in Linfinity(m). In this paper, we describe a closed ideal of A and exhibit a closed invariant subspace of H-p(m) for A that is of finite codimension. 2000 Mathematics Subject Classification.