Measurable parametrizations of sets in product spaces
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- by V. V. Srivatsa PDF
- Trans. Amer. Math. Soc. 270 (1982), 537-556 Request permission
Abstract:
Various parametrization theorems are proved. In particular the following is shown: Let $B$ be a Borel subset of $I \times I$ (where $I = [0, 1]$) with uncountable vertical sections. Let $\sum \dot \cup N$ be the discrete (topological) union of $\sum$, the space of irrationals, and $N$, the set of natural numbers with discrete topology. Then there is a map $f:I \times (\sum \dot \cup N) \to I$ measurable with respect to the product of the analytic $\sigma$-field on $I$ (that is, the smallest $\sigma$-field on $I$ containing the analytic sets) and the Borel $\sigma$-field on $\sum \dot \cup N$ such that $f(t, \cdot ): \sum \dot \cup N \to I$ is a one-one continuous map of $\sum \dot \cup N$ onto $\{ x:(t, x) \in B\}$ for each $t \in T$. This answers a question of Cenzer and Mauldin.References
- David Blackwell, On a class of probability spaces, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. II, University of California Press, Berkeley-Los Angeles, Calif., 1956, pp. 1–6. MR 0084882 J. Bourgain, Ph.D. Thesis, Univ. of Brussels, (1977).
- Douglas Cenzer and R. Daniel Mauldin, Measurable parametrizations and selections, Trans. Amer. Math. Soc. 245 (1978), 399–408. MR 511418, DOI 10.1090/S0002-9947-1978-0511418-3
- A. D. Ioffe, One-to-one Carathéodory representation theorem for multifunctions with uncountable values, Fund. Math. 109 (1980), no. 1, 19–29. MR 594322, DOI 10.4064/fm-109-1-19-29
- K. Kuratowski and A. Mostowski, Set theory, PWN—Polish Scientific Publishers, Warsaw; North-Holland Publishing Co., Amsterdam, 1968. Translated from the Polish by M. Maczyński. MR 0229526
- K. Kuratovskiĭ, Topologiya. Tom 2, Izdat. “Mir”, Moscow, 1969 (Russian). Translated from the English by M. Ja. Antonovskiĭ. MR 0259836
- R. Daniel Mauldin, Borel parametrizations, Trans. Amer. Math. Soc. 250 (1979), 223–234. MR 530052, DOI 10.1090/S0002-9947-1979-0530052-3
- R. Daniel Mauldin and H. Sarbadhikari, Continuous one-to-one parametrizations, Bull. Sci. Math. (2) 105 (1981), no. 4, 435–444 (English, with French summary). MR 640152
- Yiannis N. Moschovakis, Descriptive set theory, Studies in Logic and the Foundations of Mathematics, vol. 100, North-Holland Publishing Co., Amsterdam-New York, 1980. MR 561709
- H. Sarbadhikari and S. M. Srivastava, Parametrizations of $G_{\delta }$-valued multifunctions, Trans. Amer. Math. Soc. 258 (1980), no. 2, 457–466. MR 558184, DOI 10.1090/S0002-9947-1980-0558184-2 W. Sierpinski, Sur les images biunivoques et continues de l’ensembles de tous les nombres irrationnels, Mathematica 2 (1924), 18.
- S. M. Srivastava, Selection theorems for $G_{\delta }$-valued multifunctions, Trans. Amer. Math. Soc. 254 (1979), 283–293. MR 539919, DOI 10.1090/S0002-9947-1979-0539919-3
- John von Neumann, On rings of operators. Reduction theory, Ann. of Math. (2) 50 (1949), 401–485. MR 29101, DOI 10.2307/1969463
- Daniel H. Wagner, Survey of measurable selection theorems: an update, Measure theory, Oberwolfach 1979 (Proc. Conf., Oberwolfach, 1979) Lecture Notes in Math., vol. 794, Springer, Berlin-New York, 1980, pp. 176–219. MR 577971
- Eugene Wesley, Extensions of the measurable choice theorem by means of forcing, Israel J. Math. 14 (1973), 104–114. MR 322129, DOI 10.1007/BF02761539 W. Yankov, Sur l’uniformisation des ensembles $A$, Dokl. Akad. Nauk SSSR 30 (1941), 597-598.
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 270 (1982), 537-556
- MSC: Primary 54H05; Secondary 04A15, 28A05
- DOI: https://doi.org/10.1090/S0002-9947-1982-0645329-0
- MathSciNet review: 645329