Rocky Mountain Journal of Mathematics

Borel and continuous systems of measures

Aviv Censor and Daniele Grandini

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We study Borel systems and continuous systems of measures, with a focus on mapping properties: compositions, liftings, fibred products and disintegration. Parts of the theory we develop can be derived from known work in the literature, and in that sense this paper is of an expository nature. However, we put the above notions in the spotlight and provide a self-contained, purely measure-theoretic, detailed and thorough investigation of their properties, and in that aspect our paper enhances and complements the existing literature. Our work constitutes part of the necessary theoretical framework for categorical constructions involving measured and topological groupoids with Haar systems, a line of research we pursue in separate papers.

Article information

Rocky Mountain J. Math., Volume 44, Number 4 (2014), 1073-1110.

First available in Project Euclid: 31 October 2014

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Zentralblatt MATH identifier

Primary: 28A50: Integration and disintegration of measures 22A22: Topological groupoids (including differentiable and Lie groupoids) [See also 58H05]

System of measures Borel system of measures lifting fibred product disintegration groupoid Haar system


Censor, Aviv; Grandini, Daniele. Borel and continuous systems of measures. Rocky Mountain J. Math. 44 (2014), no. 4, 1073--1110. doi:10.1216/RMJ-2014-44-4-1073.

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  • Claire Anantharaman-Delaroche and Jean Renault, Amenable groupoids, Mono. L'Enseign. Math. 36, Geneva, 2000.
  • John C. Baez, Alexander E. Hoffnung and Christopher D. Walker, Higher-dimensional algebra VII:Groupoidification, Theor. Appl. Categ. 24 (2010), 489–553.
  • N. Bourbaki, Intégration, Chapter V, Springer, Berlin, 2007 (reprint of the 2nd edition, Hermann, Paris, 1967) (in French).
  • Aviv Censor and Daniele Grandini, Weak pullbacks of topological groupoids, New York J. Math. 18 (2012), 797–834.
  • Alain Connes, Sur la théorie non commutative de l'intégration, Lect. Notes Math. 725, Springer, Berlin, 1979 (in French).
  • Edward G. Effros, Global Structure in Von Neumann algebras, Trans. Amer. Math. Soc. 121 (1966), 434–454.
  • Raymond C. Fabec, Fundamentals of infinite dimensional representation theory, Mono. Surv. Pure Appl. Math. 114, Chapman&Hall/CRC, Boca Raton, 2000.
  • Paul S. Muhly, Coordinates in operator algebras, CBMS Lecture Notes Series, to appear.
  • John von Neumann, Zur Operatorenmethode in der klassischen Mechanik, Ann. Math. 33 (1932), 587–642 (in German).
  • Alan L.T. Paterson, Groupoids, inverse semigroups, and their operator algebras, Progr. Math. 170, Birkhauser, Boston, 1999.
  • Arlan B. Ramsay, Virtual groups and group actions, Adv. Math. 6 (1971), 253–322.
  • ––––, Polish groupoids, in Descriptive set theory and dynamical systems, Lond. Math. Soc. Lect. Note Ser. 277 (2000), 259–271.
  • Jean Renault, The ideal structure of groupoid crossed product $C^{\ast} $-algebras, J. Operator Theory 25 (1991), 3–36.
  • Jean Renault, A groupoid approach to $C^{\ast} $-algebras, Lect. Notes Math. 793, Springer, Berlin, 1980.
  • D. Revuz, Markov chains, 2nd edition, North Holland, Amsterdam, 1984.