Summary
Let A be a real or complex commutative ordered algebra with identity and involution. Let Г denote the set of positive multiplicative linear functionals ρ on A. Equip Г with the topology of simple convergence. For a fixed non-negative probability measure μ on Г the set ℒ p of linear functionals f on A which admit an integral representation of the form \(f(x) = \mathop \smallint \limits_r \rho (x)F(\rho )d\mu (\rho )\) with F∈L p (μ) (1≦p≦τ) is biuniquely identified with L p (μ) via the map tf→F. The norm on ℒ p under which this map becomes an isometry is characterized and a formula for approximating F is derived. The linear functionals which admit representation of the form \(\mathop \smallint \limits_r \rho (x)dv(\rho )\) with ν⊥μ are also characterized and appropriately normed. The theory is applied to solve abstract versions of trigonometric and n-dimensional moment problems as well as provide an alternate point of view to the theory of L p-spaces. New proofs of classical theorems are offered.
Article PDF
Similar content being viewed by others
References
Alo, R.A., de Korvin, A.: Functions of bounded Variation on idempotent semigroups. Math. Ann. 194, 1–11 (1971)
Berg, C., Christensen, J., Ressel, P.: Positive definite functions on abelian semigroups. Math. Ann. 223, 253–274 (1976)
Bochner, S.: Finitely additive integrals. Ann. of Math. 40, 769–799 (1939)
Choquet, G.: Theory of capacities. Ann. Inst. Fourier 51, 131–295 (1954)
Darst, R.B.: A decomposition of finitely additive set functions. J. für Reine und Angewandte Mathematik 210, 31–37 (1962)
Dunford, N., Schwartz, J.: Linear Operators (I). New York: Interscience Publishers 1957
Fine, N.J., Maserick, P.H.: On the simplex of completely monotonic functions of a commutative semigroup. Canad. J. Math. 22, 317–326 (1970)
Kist, J., Maserick, P.H.: BV-functions on semilattices. Pacific J. of Math. 68, 711–723 (1971)
Leader, S.: The theory of L p-spaces for finitely additive set functions Ann. of Math. 58, 528–543 (1953)
Lindahl, R.J., Maserick, P.H.: Positive definite functions on semigroups. Duke Math. J. 38, 711–782 (1971)
Lorentz, G.: Bernstein Polynomials. Math. Expositions no. 8. Toronto: Univ. of Toronto Press 1953
Maserick, P.H.: Moment and BV-functions on commutative semigroups. Trans. Amer. Math. Soc. 181, 61–75 (1973)
Maserick, P.H.: BV-functions, positive-definite functions and moment problems. Trans. Amer. Math. Soc. 214, 137–152 (1975)
Maserick, P.H.: Moments of measures on convex bodies, Pacific J. of Math. 68, 135–152 (1977)
Maserick, P.H.: Désintégration à densité L p et mesures singulières. C.R. Acad. Sci. Paris, 289, 63–64 (1979)
Maserick, P.H.: Differentiation of ℒ p-functions on semilattices, to appear in the Pacific J. of Math.
Newman, S.E.: Measure algebras and functions of bounded variation on idempotent semigroups. Trans. Amer. Math. Soc. 163, 189–205 (1972)
Rudin, W.: Fourier Analysis on Groups. New York-London: Interscience Publishess 1962
Schoenberg, I.J.: A remark on a preceding note by Bochner. Bull. Amer. Math. Soc. 40, 277–278 (1934)
Stewart, J.: Fourier Transforms of Unbounded Measures. Canad. J. Math. 31, 1281–1292 (1979)
Widder, D.V.: The Laplace Transforms. Princeton: Princeton Univ. Press. 1941
Author information
Authors and Affiliations
Additional information
Research for this paper was sponsored in part by the Danish Natural Science Research Council (Grant No.511-10302) and in part by the National Science Foundation (Grant No. MCS78-03397)
The results contained herein include the proofs of theorems announced in [15]
Rights and permissions
About this article
Cite this article
Maserick, P.H. Disintegration with respect to L p-density functions and singular measures. Z. Wahrscheinlichkeitstheorie verw Gebiete 57, 311–326 (1981). https://doi.org/10.1007/BF00534826
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00534826