Vol. 15, No. 2, 1965

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 329: 1  2
Vol. 328: 1  2
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Vol. 324: 1  2
Vol. 323: 1  2
Vol. 322: 1  2
Online Archive
The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author index
To appear
Other MSP journals
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