Abstract.
In this paper we prove, for signed or complex Radon measures on completely regular spaces, the analogue of Prokhorov's criterion on the existence of the projective limit of a compatible system of measures. Because of loss of mass under projections this cannot be reduced to the case of positive measures. Countable projective limits are, as in the case of positive measures, particularly simple, the sole condition now being the boundedness of the total variations. It is shown, with the help of the martingale convergence theorem, that the densities of these complex measures with respect to their variations, converge in an appropriate sense. This work is part of an extended project on the mathematical theory of path integrals.
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Received: 18 June 1997 / Revised version: 11 August 2000/¶Published online: 9 March 2001
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Thomas, E. Projective limits of complex measures and martingale convergence. Probab Theory Relat Fields 119, 579–588 (2001). https://doi.org/10.1007/PL00008772
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DOI: https://doi.org/10.1007/PL00008772
- Mathematics Subject Classification (2000): Primary 28A10, 28A35, Secondary 81S40
- Key words or phrases: Complex Radon measure – Projective limit