Vol. 15, No. 2, 1965

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On the generalized F. and M. Riesz theorem

Patrick Robert Ahern

Vol. 15 (1965), No. 2, 373–376

Let X be a compact Hausdorff space, C(X) the algebra of all continuous complex valued functions on X, and let A be a sup-norm algebra on X, that is, A is a uniformly closed algebra of continuous complex valued functions on X that contains the constants and separates the points. If ϕ is a complex homomorphism of A then let M(ϕ) be the set of all positive, regular, Borel measures on X that represent ϕ. If μ is a finite, (complex), regular, Borel measure on X then we write μ A if f dμ = 0 for all f A. Let ϕ be a complex homomorphism of A and m M(ϕ), then we say that m satisfies the Riesz theorem if whenever μ is a finite, (complex), regular, Borel measure on X and μ A then μa A and μs A where μ = μa + μs is the Lebesgue decomposition of μ with respect to m. It is quite easy to see that if m M(ϕ) and m satisfies the Riesz theorem then for all ρ M(ϕ) we have ρ is absolutely continuous with respect to m. We will show that this condition is also sufficient. This is done by means of a theorem which says that if F X is a compact Gδ such that m(F) = 0 for all m M(ϕ) then there exists a sequence fn in A such that |fn|1 on X, ϕ(fn) 1, and fn 0 uniformly on F.

Mathematical Subject Classification
Primary: 46.55
Secondary: 46.25
Received: 7 April 1964
Published: 1 June 1965
Patrick Robert Ahern
Department of Mathematics
University of Wisconsin
Madison WI
United States