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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

On the continuity of Haar measure on topological groupoids


Author: Anthony Karel Seda
Journal: Proc. Amer. Math. Soc. 96 (1986), 115-120
MSC: Primary 46L99; Secondary 22D40, 28C10, 43A05
MathSciNet review: 813822
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Abstract: It is shown that continuity of a family of invariant (Haar) measures on a topological groupoid $ G$ is equivalent to the continuity of the implied convolution product $ f * g$ for all pairs of functions $ f$ and $ g$. An example is given of a groupoid which admits no (continuous) Haar measure. It results, therefore, that the usual $ {C^ * }$-algebra associated with a Haar measure on $ G$ cannot, in general, be constructed. Some remarks are included concerning the construction of Haar measures on the holonomy groupoid of a foliated manifold.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1986-0813822-2
PII: S 0002-9939(1986)0813822-2
Article copyright: © Copyright 1986 American Mathematical Society