Abstract
Let G be a connected locally compact abelian group and ν a symmetric Gaussian measure on G. We are concerned with the support of the measure ν and with the relation of ν to the Haar measure ω on G. It is shown that the support of ν is always a closed connected subgroup of G. On G there exists an absolutely continuous Gaussian measure (with respect to ω) if and only if G is locally connected and has a countable basis for its open sets. Special interest is given to Gaussian measures on toroidal groups.
Similar content being viewed by others
Literatur
DIXMIER, J.: Quelques propriétés des groupes abéliens localement compacts. Bull. Sci. math., II. Sér.81, 38–48 (1957).
GREUB, W.H.: Linear Algebra, 3.Ed. Berlin-Heidelberg-New York: Springer 1967.
HEWITT, E., ROSS, K.A.: Abstract Harmonic Analysis I, II. Berlin-Heidelberg-New York: Springer 1963/70
HEWITT, E., STROMBERG, K.: Real and Abstract Analysis. Berlin-Heidelberg-New York: Springer 1969.
HEYER, H.: Fourier transforms and probabilities on locally compact groups. Jahresbericht der DMV70, 109–147 (1968).
HEYER, H., RALL, Chr.: Gaußsche Wahrscheinlichkeitsmaße auf Corwinschen Gruppen. Math.Z.128, 343–361 (1972).
LINNIK, Y.-V.: Décompositions des lois de probabilités. Gauthier-Villars: Paris 1962.
PONTRJAGIN, L.S.: Topologische Gruppen, Teil 2, 2. Aufl. Leipzig: Teubner 1958.
RICKERT, N.W.: Some properties of locally compact groups. J.Austral.math.Soc.VII, 433–454 (1967).
SIEBERT, E.: Stetige Halbgruppen von Wahrscheinlichkeitsmaßen auf lokalkompakten maximal fastperiodischen Gruppen. Z.Wahrscheinlichkeitstheorie verw. Gebiete25, 269–300 (1973).
SIEBERT, E.: Absolut-Stetigkeit und Träger von Gauß-Verteilungen auf lokalkompakten Gruppen. Math. Ann.210, 129–147 (1974)
URBANIK, K.: Gaussian measures on locally compact abelian topological groups. Studia math. Ser.spec.XIX, 77–88 (1960).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Siebert, E. Einige Bemerkungen zu den Gauss-Verteilungen auf lokalkompakten abelschen Gruppen. Manuscripta Math 14, 41–55 (1974). https://doi.org/10.1007/BF01637621
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01637621