**Abstract:** Let G be an infinite compact abelian group,
mu a Borel measure on G with spectrum E, and 0 < p < 1.
We show that if mu is not absolutely continuous with respect
to Haar measure, then L^p_E(G), the closure in L^p(G) of the
E-trigonometric polynomials, does not have enough continuous
linear functionals to separate points. If mu is actually singular,
then L^p_E(G) does not have any nontrivial continuous linear
functionals at all. Our methods recover the classical F. and
M. Riesz theorem, and a related several variable result of Bochner;
they reveal the existence of small sets of characters that span
L^p(T), where T is the unit circle; and they show that, for 0<p<1,
the H^p spaces of the "big disc algebra" have one-dimensional
dual. |