Abstract
In Macheras and Strauss (Atti Sem Math Fis Univ Modena, L, pp 349–361, 2002) and Musial et al. (J Theor Probab 20:545–560, 2007) various products for primitive liftings in the factors of a product of probability spaces have been considered. In this paper we settle for the d-dimensional Lebesgue densities open problems from Macheras and Strauss (Atti Sem Math Fis Univ Modena, L, pp 349–361, 2002) and Musial et al. (J Theor Probab 20:545–560, 2007) by applying results relying on the metrical group structure of \({{\mathbb R}^d}\), if \({d\in{\mathbb N}}\). In particular, a lifting problem from Musial et al. (Arch Math 83:467–480, 2004), Question 3.3, is decided to the negative for the Lebesgue densities. The relation of the Lebesgue density in the product space and the results of the products taken for the Lebesgue densities in the factors under order is discussed. The results can be carried over to densities and liftings dominating Lebesgue densities and to multiplicative and positive linear liftings on function spaces.
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References
Balcerzak, M., Hejduk, J.: Density topologies for products of σ-ideals. Real Anal. Exchange 20(1), 163–177 (1994/1995)
Burke M.R., Macheras N.D., Musiał K., Strauss W.: Category product densities and liftings. Topol. Appl. 153, 1164–1191 (2006)
Burke M.R., Macheras N.D., Musiał K., Strauss W.: Various products of category densities and liftings. Topol. Appl. 156, 1253–1270 (2009)
Cenzer D., Mauldin R.D.: Measurable parametrizations and selections. Trans. Am. Math. Soc. 245, 399–408 (1978)
Erdös P., Oxtoby J.C.: Partitions of the plane into sets having positive measure in every non-null product set. Trans. Am. Math. Soc. 79, 91–102 (1955)
de Guzman, M.: Differentiation of Integrals. Lecture Notes in Mathematics, vol. 481. Springer, Berlin (1975)
Ionescu Tulcea, A., Ionescu Tulcea, C.: Topics in the Theory of Lifting. Springer, Berlin (1969)
Macheras N.D., Strauss W.: On products of almost strong liftings. J. Austral. Math. Soc. (Series A) 60, 311–333 (1996)
Macheras, N.D., Strauss, W.: One-sided Fubini formulas. Atti Sem. Math. Fis. Univ. Modena, L, pp. 349–361 (2002)
Macheras, N.D., Strauss, W.: Lifting topologies for primitive liftings. Atti Sem. Mat. Fis. Univ. Modena e Regio Emilia, LIII, 345–363 (2005)
Mauldin D.: One-to-one selections-marriage theorems. Am. J. Math. 104(4), 823–828 (1982)
Musiał K., Strauss W., Macheras N.D.: Product liftings and densities with lifting invariant and density invariant sections. Fund. Math. 166, 281–303 (2000)
Musiał K., Strauss W., Macheras N.D.: Existence of linear liftings with invariant sections in product measure spaces. Arch. Math. 83, 467–480 (2004)
Musiał K., Strauss W., Macheras N.D.: Fubini type products for densities and liftings. J. Theor. Probab. 20, 545–560 (2007)
Saks S.: Theory of the Integral. G. E. Stechert, New York (1937)
Strauss, W., Macheras, N.D., Musiał, K.: Liftings. In: Handbook of Measure Theory, pp. 1131–1184. Elsevier, Amsterdam (2002)
Wagner D.H.: Survey of measurable selection theorems. SIAM J. Control Optim. 15(5), 859–903 (1977)
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Macheras, N.D., Strauss, W. Various products for Lebesgue densities. Positivity 14, 815–829 (2010). https://doi.org/10.1007/s11117-010-0084-6
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DOI: https://doi.org/10.1007/s11117-010-0084-6