Random selection of Borel sets
Keywords:Random Borel sets, Dyadic spaces, , Sierpinski’s universal curve
A theory of random Borel sets is presented, based on dyadic resolutions of compact metric spaces. The conditional expectation of the intersection of two independent random Borel sets is investigated. An example based on an embedding of Sierpinski’s universal curve into the space of Borel sets is given.
P. Alexandroff and H. Hopf, Topologie, Chelsea, 1972.
N. Bourbaki, General Topology, Elements of Mathematics, Hermann and Addison-Wesley, 1966.
N. Bourbaki, Topological Vector Spaces I-V, Elements of Mathematics, Springer, 1987.
N. Bourbaki, Integration I (Chapters 1-6), Elements of Mathematics, Springer, 2004.
P. R. Halmos,Measure Theory, volume 18 of GTM, Springer, 1974.
F. Hausdorff, Mengenlehre, G¨oschens Lehrbücherei. Walter de Gruyter & Co., 2nd edition, 1927.
S. B. Nadler Jr, Continuum Theory, volume 158 of Pure and Applied Mathematics, Marcel Dekker, Inc., 1992.
A. S. Kechris, Classical descriptive set theory, volume 156 of GTM, Springer, 1995.
S. Li, Y. Ogura and V. Kreinovich, Limit Theorems and Applications of Set-Valued and Fuzzy Set-Valued Random Variables, Theory and Decisions Library, Kluwer, 2002. http://dx.doi.org/10.1007/978-94-015-9932-0
J. Lindenstrauss, A short proof of liapounoff’s convexity theorem, J. Math. Mech. 15 (1966), no. 6, 971–972.
I. Molchanov, Theory of random sets, Probability and Its Applications. Springer, 2005.
H. E. Robbins, On the measure of random set, Ann. Math. Statist. 15 (1944), 70–74. http://dx.doi.org/10.1214/aoms/1177731315
H. E. Robbins, On the measure of random set II, Ann. Math. Statist. 16 (1945), 342–347. http://dx.doi.org/10.1214/aoms/1177731060
C. Rosendal, The generic isometry and measure preserving homeomorphism are conjugate to their powers, Fund. Math. 205 (2009), 1–27. http://dx.doi.org/10.4064/fm205-1-1
R. Schneider and W. Weil, Stochastic and Integral Geometry, Probability and its Applications, Springer, 2008. http://dx.doi.org/10.1007/978-3-540-78859-1
W. Sierpi´nski, Sur les fonctions d’ensemble additives et continues, Fund. Math. 3 (1922), 240–246.
F. Straka and J. Stepán, Random sets in [0,1], In J. Visek and S. Kubik, editors, Information theory, statistical decision functions, random processes, Prague 1986, volume B, pages 349–356, Reidel, 1989.
P. Wojtaszczyk, A Mathematical Introduction to Wavelets, volume 37 of Student Text, London Mathematical Society, 1997.
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