A Generalization of the Rudin-carleson Theorem

The purpose of this paper is to prove a common generalization of a theorem due to T. W. Gamelin [3] and a theorem due to Z. Semadeni [5]. Both these results are generalizations of E. Bishop's abstract version of the well-known RudinCarleson theorem [2]. In the following X denotes a compact Hausdorff space, F a closed subset of X and C(A') and C(F) denote the spaces of all complex-valued functions on the topological spaces X and F respectively. A denotes a closed linear subspace of C(A") with respect to the sup norm topology, and A\F denotes the set of all restrictions of the elements of A to F. For/e A,f\F denotes the restriction off to F and for p e M(X), the set of all complex Radon measures on X, pF denotes the restriction of p to F. By A L we understand the set of all elements p e M(X) with the property that 5X fdp=0 for allfeA. Our purpose in this paper is to present a common generalization of the following two theorems : Theorem 1 (Semadeni). Assume that the condition, (*) p e A L => pF = 0 for all p e M(X), is satisfied. Let a0 e C(F) and let ip:X—>(0, co] be a lower semicontinuous function such that \a0(x)\~¿xp(x) for all x e F. Then there exists an ä e A such that (î|F=a0 and \ä(x)\^y>(x)for all xeX. Theorem 2 (Gamelin). Assume that the condition, (**) p e A1 => pF e A1for all p e M(X), is satisfied. Let a0eA\F and let p:X—>(0, oo) be a continuous function such that \a0(x)\^p(x) for all x e F. Then there exists an ä e A such that ä\F=a0 and \ä(x)\^p(x) for all x e X. Our theorem is the following: Theorem 3. Assume that condition (**) is satisfied. Let a0 e A\F and let ip:X-+(0, oo] be a lower semicontinuous function such that \a0(x)\^xp(x) Received by the editors March 19, 1973. AMS (MOS) subject classifications (1970). Primary 46J10; Secondary 54C20. © American Mathematical Society 1974 341 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use