Real Analysis Exchange

Weak Convergence of Bounded, Monotone Set Functions in an Abstract Setting

Bruno Girotto and Silvano Holzer

Full-text: Open access

Abstract

We introduce an abstract treatment of the weak convergence for bounded monotone set functions which allows us to obtain some basic results generalizing well known theorems regarding classical weak and vague convergence and weak convergence of masses on normal topological spaces (e.g. Portmanteau type theorems, Direct and Converse Prokhorov type theorems). Moreover, we introduce a suitable topology (called the L\'evy-topology) in order to study the properties of this abstract convergence from a topological point of view.\newline

Article information

Source
Real Anal. Exchange, Volume 26, Number 1 (2000), 157-176.

Dates
First available in Project Euclid: 2 January 2009

Permanent link to this document
https://projecteuclid.org/euclid.rae/1230939151

Mathematical Reviews number (MathSciNet)
MR1825501

Zentralblatt MATH identifier
1014.28505

Subjects
Primary: Primary28A12 28A33: Spaces of measures, convergence of measures [See also 46E27, 60Bxx] Secondary60B10 60B05: Probability measures on topological spaces

Keywords
Monotone set function Choquet integral weak convergence L\'evy-topology tightness

Citation

Girotto, Bruno; Holzer, Silvano. Weak Convergence of Bounded, Monotone Set Functions in an Abstract Setting. Real Anal. Exchange 26 (2000), no. 1, 157--176. https://projecteuclid.org/euclid.rae/1230939151


Export citation

References

  • Alexandroff, D.(1940-43). Additive set functions in abstract spaces, Mat. Sb. 8, 307-348; 9, 563-628; 13, 169-238.
  • Bhaskara Rao, K.P.S. and Bhaskara Rao, M.(1983). Theory of Charges, Academic Press, New York.
  • Billingsley, P.(1968). Convergence of probability measures, John Wiley, New York.
  • Denneberg, D.(1994). Non-additive measure and integral, Kluwer Academic Publisher, Dordrecht.
  • Engelking, R.(1977). General topology, PWN-Polish Scientific Publishers, Warszawa.
  • Girotto, B. and Holzer, S.(1993). Weak convergence of masses on normal topological spaces, Theory Probab. Appl. 37, 337-340 (an English translation of the June 1992 issue of the Soviet journal Teoriya Veroyatnostei i ee Primeneiya, 364-366).
  • Girotto, B. and Holzer, S.(1993). Weak convergence of masses on normal topological spaces, Sankhya A 55, 188-201.
  • Girotto, B. and Holzer, S.(2000). Weak convergence of masses: topological properties, Atti Sem. Mat. Fis. Univ. Modena 48(2), to appear.
  • Masani, P.(1982). The outer regularization of finitely-additive measures over normal topological spaces, Proc. Measure Theory Conf., Oberwolfach, 1981, Lecture Notes in Math. 945, 116-144.
  • Varadarajan, V.S.(1958). Weak convergence of measures on separable metric spaces, Sankhya 19, 15-22.