Counting measures, monotone random set functions

  • T. E. Harris
Article
Received:

Summary

This paper arose from work on random processes whose values are measures or more general set functions. Secs. 1–3, which have nothing specifically “random”, discuss two topologies for certain sigma-finite measures. One, applicable only to counting measures, is a quotient topology which is useful in the finite case but excessively weak in the infinite case. Making use of a well-known result of P. Hall on sets of representatives, we describe this topology and show that it can be enlarged to the stronger one generated by a modification of the Lévy-Prohorov (L-P) metric. Sec. 4 gives a property of the L-P metric for finite integer valued counting measures. The rest of the paper deals with a random monotone non-negative set function Ω in a separable metric space X. If X is complete and if Ω is subadditive and right continuous1 in probability on certain classes of sets, we show the existence of a version of Ω with right-continuous sample functions. If X is locally compact and Ω is left continuous in probability on a certain class of open sets, there is a left-continuous version. With appropriate additional assumptions, we obtain versions that are measures or capacities. In the latter case, a 0–1 valued set function represents a random closed or compact set. The form of integer-valued strongly subadditive set functions is described for certain cases.

Keywords

Random Process Mathematical Biology Counting Measure Sample Function Quotient Topology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bochner, S.: Harmonic analysis and the theory of probability. University of California, 1960.Google Scholar
  2. 2.
    Choquet, G.: Theory of capacities. Ann. Inst. Fourier 5, 131–295 (1955).Google Scholar
  3. 3.
    Dold, A., u. R. Thom: Quasifaserungen und unendliche symmetrische Produkte. Ann. of Math. II. Ser. 67, 239–281 (1958).Google Scholar
  4. 4.
    Doob, J. L.: Stochastic processes. London-New York: Wiley 1953.Google Scholar
  5. 5.
    Dynkin, E. B.: Markov processes. Berlin-Heidelberg-New York: Springer 1965.Google Scholar
  6. 6.
    FordJr., L. R., and D. R. Fulkerson. flows in networks. Princeton: Princeton University Press 1962.Google Scholar
  7. 7.
    Grenander, U.: Probabilities on algebraic structures. London-New York: Wiley 1963.Google Scholar
  8. 8.
    Hall, P.: On representatives of subsets. J. London math. Soc. 10, 26–30 (1935).Google Scholar
  9. 9.
    Harris, T. E.: The theory of branching processes. Berlin-Göttingen-Heidelberg: Springer 1963.Google Scholar
  10. 10.
    —: Infinite product Markov processes. Trans. Amer. math. Soc. 130, 141–152 (1968).Google Scholar
  11. 11.
    Hewitt, E., and K. Stromberg: Real and abstract analysis. Berlin-Heidelberg- New York: Springer 1965.Google Scholar
  12. 12.
    Ikeda, N., M. Nagasawa, S. Watanabe: On branching Markov processes. Proc. Japan Acad. 41, 816–821 (1965).Google Scholar
  13. 13.
    Kingman, J. F. C.: Completely random measures. Pacific J. Math. 21, 59–78 (1967).Google Scholar
  14. 14.
    Kuratowski, C.: Sur une généralisation de la notion d'homomorphie. Fundamenta Math. 22, 206–220 (1934).Google Scholar
  15. 15.
    LeCam, L.: A stochastic description of precipitation. Proc. Fourth Berkeley Sympos. math. Statist. Probability 3, 165–186 (1961).Google Scholar
  16. 16.
    Lee, P. M.: Infinitely divisible stochastic processes. Z. Wahrscheinlichkeitstheorie verw. Geb. 7, 147–160 (1967).Google Scholar
  17. 17.
    Lévy, P.: Remarks sur certains ensembles aléatoires. J. Math. pur. appl., IX. Sér. 39, 97–118 (1960).Google Scholar
  18. 18.
    Meyer, P. A.: Probability and potentials. New York: Blaisdell 1966.Google Scholar
  19. 19.
    Moyal, J. E.: The general theory of population processes. Acta math. 108, 1–31 (1962).Google Scholar
  20. 20.
    Nelson, E.: Regular probability measures on function space. Ann. of Math., II. Ser. 69, 630–643 (1959).Google Scholar
  21. 21.
    Prohorov, YU. V.: Convergence of random processes and limit theorems in probability theory. Theor. Probab. Appl. 1, 157–214 (1956).Google Scholar
  22. 22.
    —: On random measures on compacts (Russian). Doklady Akad. Nauk SSSR, n. Ser. 138, 53–55 (1961).Google Scholar
  23. 23.
    Sierpinski, W.: General Topology, second Edition. Toronto: University of Toronto Press 1956.Google Scholar
  24. 24.
    Strassen, V.: The existence of probability measures with given marginals. Ann. math. Statistics 36, 423–439 (1965).Google Scholar
  25. 25.
    Varadarajan, V. S.: Measures on topological spaces (Russian). Mat. Sbornik, n. Ser. 55, 35–100 (1961).Google Scholar

Copyright information

© Springer-Verlag 1968

Authors and Affiliations

  • T. E. Harris
    • 1
  1. 1.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA

Personalised recommendations