The Annals of Probability

A strong law of large numbers for capacities

Fabio Maccheroni and Massimo Marinacci

Full-text: Open access


We consider a totally monotone capacity on a Polish space and a sequence of bounded p.i.i.d. random variables. We show that, on a full set, any cluster point of empirical averages lies between the lower and the upper Choquet integrals of the random variables, provided either the random variables or the capacity are continuous.

Article information

Ann. Probab. Volume 33, Number 3 (2005), 1171-1178.

First available in Project Euclid: 6 May 2005

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 28A12: Contents, measures, outer measures, capacities 60F15: Strong theorems

Capacities Choquet integral strong law of large numbers contents measures outer measures strong theorems


Maccheroni, Fabio; Marinacci, Massimo. A strong law of large numbers for capacities. Ann. Probab. 33 (2005), no. 3, 1171--1178. doi:10.1214/009117904000001062.

Export citation


  • Artstein, Z. and Vitale, R. A. (1975). A strong law of large numbers for random compact sets. Ann. Probab. 3 879--882.
  • Aumann, R. J. (1965). Integrals of set-valued functions. J. Math. Anal. Appl. 12 1--12.
  • Castaldo, A., Maccheroni, F. and Marinacci, M. (2004). Random correspondences as bundles of random variables. Sankhyā 66 409--427.
  • Choquet, G. (1954). Theory of capacities. Ann. Inst. Fourier 5 131--295.
  • Colubi, A., López-Díaz, M., Domínguez-Menchero, J. S. and Gil, M. A. (1999). A generalized strong law of large numbers. Probab. Theory Related Fields 114 401--417.
  • Dellacherie, C. and Meyer, P.-A. (1978). Probabilities and Potential. North-Holland, Amsterdam.
  • Dempster, A. (1967). Upper and lower probabilities induced by a multivalued mapping. Ann. Math. Statist. 38 325--339.
  • Doob, J. (1984). Classical Potential Theory and Its Probabilistic Counterpart. Springer, Berlin.
  • Epstein, L. and Schneider, M. (2003). IID: Independently and indistinguishably distributed. J. Econom. Theory 113 32--50.
  • Hess, C. (1984). Quelques théorèmes limites pour des ensembles aléatoires bornés ou non. In Séminaire d'Analyse Convexe 14. Univ. Sciences et Techniques de Langvedoc, Montpelier.
  • Huber, P. J. and Strassen, V. (1973). Minimax tests and the Neyman--Pearson lemma for capacities. Ann. Statist. 1 251--263.
  • Klein, E. and Thompson, A. C. (1984). Theory of Correspondences. Wiley, New York.
  • Marinacci, M. (1999). Limit laws for non-additive probabilities and their frequentist interpretation. J. Econom. Theory 84 145--195.
  • Marinacci, M. and Montrucchio, L. (2004). Introduction to the mathematics of ambiguity. In Uncertainty in Economic Theory (I. Gilboa, ed.) 46--107. Routledge, New York.
  • Nguyen, H. T. (1978). On random sets and belief functions. J. Math. Anal. Appl. 65 531--542.
  • Philippe, F., Debs, G. and Jaffray, J.-Y. (1999). Decision making with monotone lower probabilities of infinite order. Math. Oper. Res. 24 767--784.
  • Proske, F. N. and Puri, M. L. (2003). A strong law of large numbers for generalized random sets from the viewpoint of empirical processes. Proc. Amer. Math. Soc. 131 2937--2944.
  • Schmeidler, D. (1972). Cores of exact games. J. Math. Anal. Appl. 40 214--225.
  • Schmeidler, D. (1989). Subjective probability and expected utility without additivity. Econometrica 57 571--587.
  • Shafer, G. (1976). A Mathematical Theory of Evidence. Princeton Univ. Press.
  • Srivastava, S. M. (1998). A Course on Borel Sets. Springer, New York.