Hereditary properties of the class of closed sets of uniqueness for trigonometric series

It is shown that theσ-ideal U_0 of closed sets of extended uniqueness in T is hereditarily non-Borel, i.e. every "non-trivial" σ-ideal of closed sets I ⊆ U_0 is non-Borel. This implies both the result of Solovay, Kaufman that both U_0 and U(the σ-ideal of closed sets of uniqueness) are not Borel as well as the theorem of Debs-Saint Raymond that every Borel subset of T of extended uniqueness is of the first category. A further extension to ideals contained in U_0 is given