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We generalize the classical F. and M. Riesz theorem to metrizable compact groups whose center contains a copy of the circle group. Important examples of such groups are the isotropy groups of the bounded symmetric domains. The proof uses a criterion for absolute continuity involving $L^p$ spaces with $p\<1$: A measure $\mu$ on a compact metrisable group $K$ is absolutely continuous with respect to Haar measure $dk$ on $K$ if for some $p\<1$ a certain subspace of $L^p(K,dk)$ which is related to $\mu$ has sufficiently many...