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Trace measures of a positive definite bimeasure

Author

Listed:
  • Dehay, Dominique
  • Moché, Raymond

Abstract

If a positive definite bimeasure, M: - --> C, on any measurable space ([Omega], ) is extendable to a complex measure M', on the product [sigma]-algebra [circle times operator] , one can define its trace M'[Delta] on the diagonal [Delta] of [Omega] - [Omega]. If M is not extendable, the definition of M'[Delta] fails, but the need for defining some measures on [Delta] which would act as M'[Delta] remains, especially in the spectral analysis and the analysis of the stationarity of weakly harmonizable processes. In this paper, we fulfill this need with two types of measures concentrated on [Delta] or, more generally, on simple curves of [Omega] - [Omega]. The first ones, which have been called quasi-trace measures, are constructed under the assumption that the given positive definite bimeasure M is [sigma]-finite. They have the expected properties. The second ones, which are called pseudo-trace measures, are constructed, whatever the positive definite bimeasure M be, but they very often lead to new problems.

Suggested Citation

  • Dehay, Dominique & Moché, Raymond, 1992. "Trace measures of a positive definite bimeasure," Journal of Multivariate Analysis, Elsevier, vol. 40(1), pages 115-131, January.
    • Handle: RePEc:eee:jmvana:v:40:y:1992:i:1:p:115-131
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