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Then μ is absolutely continuous with respect to Lebesgue measure on T. From this theorem, we can deduce the famous Rudin–Carleson theorem: Let F be a closed subset of T. Every continuous function on F can be extended to an analytic function inside the unit disk with boundary value on F equal to the given continuous function on F if and only if F is a Lebesgue measure zero set. Later, Helson and Lowdenslager generalized the F. and M. Riesz theorem to compact Abelian groups with ordered duals. De Leeuw and Glicksberg, Doss and Yamaguchi shortly afterwards obtained a number of related results.

Recent developments on the F. and M. Riesz theorem