Journal of Theoretical Probability

, Volume 20, Issue 3, pp 545–560 | Cite as

Fubini Type Products for Densities and Liftings



In our former paper (Fund. Math. 166, 281–303, 2000) we discussed densities and liftings in the product of two probability spaces with good section properties analogous to that for measures and measurable sets in the Fubini Theorem. In the present paper we investigate the following more delicate problem: Let (Ω,Σ,μ) and (Θ,T,ν) be two probability spaces endowed with densities υ and τ, respectively. Can we define a density on the product space by means of a Fubini type formula \((\upsilon\odot\tau)(E)=\{(\omega,\theta):\omega\in\upsilon(\{\bar {\omega}:\theta\in\tau(E_{\bar{\omega}}\})\}\) , for E measurable in the product, and the same for liftings instead of densities? We single out classes of marginal densities υ and τ which admit a positive solution in case of densities, where we have sometimes to replace the Fubini type product by its upper hull, which we call box product. For liftings the answer is in general negative, but our analysis of the above problem leads to a new method, which allows us to find a positive solution. In this way we solved one of the main problems of Musiał, Strauss and Macheras (Fund. Math. 166, 281–303, 2000).


Product densities Product liftings Product probability spaces 


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© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of MathematicsWrocław University, Institute of MathematicsWrocławPoland
  2. 2.Universität Stuttgart, Fachbereich Mathematik Institut für Stochastik und Anwendungen Abteilung für Finanz- und VersicherungsmathematikStuttgartGermany
  3. 3.Department of Statistics and Insurance ScienceUniversity of PiraeusPiraeusGreece

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