Journal of Theoretical Probability

, Volume 20, Issue 3, pp 545–560

# Fubini Type Products for Densities and Liftings

• K. Musiał
• W. Strauss
• N. D. Macheras
Article

## Abstract

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).

## Keywords

Product densities Product liftings Product probability spaces

## References

1. 1.
Fremlin, D.H.: Measure Theory, vol. 3. T. Fremlin (ed.) (2002) Google Scholar
2. 2.
Gapaillard, J.: Relevements monotones. Arch. Math. 24, 169–178 (1973)
3. 3.
Graf, S., von Weizsäcker, H.: On the existence of lower densities in non-complete measure spaces. In: Bellow, A., Kölzow, D. (eds.) Measure Theory (Oberwolfach 1975). Lecture Notes in Math., vol. 541, pp. 155–158. Springer, Berlin (1976) Google Scholar
4. 4.
Ionescu Tulcea, A., Ionescu Tulcea, C.: Topics in the Theory of Lifting. Springer, Berlin (1969)
5. 5.
Macheras, N.D., Musiał, K., Strauss, W.: On products of admissible liftings and densities. Z. Anal. Anw. 18, 651–667 (1999)
6. 6.
Macheras, N.D., Musiał, K., Strauss, W.: Linear liftings respecting coordinates. Adv. Math. 153, 403–416 (2000)
7. 7.
Macheras, N.D., Musiał, K., Strauss, W.: On strong liftings on projective limits. Glasgow Math. J. 45, 503–525 (2003)
8. 8.
Macheras, N.D., Strauss, W.: Products of lower densities. Z. Anal. Anw. 14, 25–32 (1995)
9. 9.
Macheras, N.D., Strauss, W.: On products of almost strong liftings. J. Austr. Math. Soc. (Ser. A) 60, 311–333 (1996)
10. 10.
Musiał, K., Strauss, W., Macheras, N.D.: Product liftings and densities with lifting invariant and density invariant sections. Fund. Math. 166, 281–303 (2000) Google Scholar
11. 11.
Musiał, K., Strauss, W., Macheras, N.D.: Liftings for topological products of measures. In: Atti Sem, Mat. Fis. Univ. Modena e Reggio Emilia, LII, pp. 69–72 (2004) Google Scholar
12. 12.
Strauss, W., Macheras, N.D., Musiał, K.: Liftings. In: Handbook of Measure Theory, pp. 1131–1184. Elsevier, Amsterdam (2002) Google Scholar
13. 13.
Strauss, W., Macheras, N.D., Musiał, K.: Non-existence of certain types of liftings and densities in product spaces with σ-ideals. Real Anal. Exch. 29(1), 473–479 (2003/2004) Google Scholar
14. 14.
Strauss, W., Macheras, N.D., Musiał, K.: Splitting of liftings in probability spaces. Ann. Probab. 32, 2389–2408 (2004)
15. 15.
Talagrand, M.: Closed convex hull of set of measurable functions, Riemann-measurable functions and measurability of translations. Ann. Inst. Fourier (Grenoble) 32, 39–69 (1989) Google Scholar