Measurable parametrizations and selections
Authors:
Douglas Cenzer and R. Daniel Mauldin
Journal:
Trans. Amer. Math. Soc. 245 (1978), 399408
MSC:
Primary 28A20; Secondary 04A15, 54H05
MathSciNet review:
511418
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Abstract: Let W be a Borel subset of (where ) such that, for each x, is uncountable. It is shown that there is a map, g, of onto W such that (1) for each x, is a Borel isomorphism of I onto and (2) both g and are measurable maps. Here, if X is a topological space, is the smallest family containing the open subsets of X which is closed under operation (A) and complementation. Notice that is a subfamily of the universally or absolutely measurable subsets of X. This result answers a problem of A. H. Stone. This result improves a theorem of Wesley and as a corollary a selection theorem is obtained which extends the measurable selection theorem of von Neumann. We also show an analogous result holds if W is only assumed to be analytic.
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 D. Cenzer and R. D. Mauldin, Inductive definability, measure and category (to appear). MR 594994 (82b:03086)
 [2]
 J. R. Choksi, Measurable transformations on compact groups, Trans. Amer. Math. Soc. 184 (1973), 101124. MR 0338311 (49:3076)
 [3]
 F. Hausdorff, Set theory, Chelsea, New York, 1964.
 [4]
 K. Kunugui, Sur un théorème d'existence dans la théorie des ensembles projectifs, Fund. Math. 29 (1937), 169181.
 [5]
 K. Kuratowski, Topology, Vol. I, Academic Press, New York, 1966. MR 0217751 (36:840)
 [6]
 R. D. Mauldin, Borel parameterizations (preprint).
 [7]
 J. von Neumann, On rings of operators; reduction theory, Ann. of Math. 30 (1949), 401485. MR 0029101 (10:548a)
 [8]
 P. S. Novikov, Sur les projections de certains ensembles mesurables B, Dokl. Akad. Nauk. SSSR (N.S.) 23 (1939), 864865.
 [9]
 G. E. Sacks, Measuretheoretic uniformity, Trans. Amer. Math. Soc. 142 (1969), 381420. MR 0253895 (40:7108)
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 A. H. Stone, Measure theory, Lecture Notes in Math., vol. 541, SpringerVerlag, Berlin and New York, 1976, pp. 4348. MR 0450492 (56:8786)
 [11]
 D. H. Wagner, Survey of measurable selection theorems, SIAM J. Control Optimization 15 (1977), 859903. MR 0486391 (58:6137)
 [12]
 E. Wesley, Extensions of the measurable choice theorem by means of forcing, Israel J. Math. 14 (1973), 104114. MR 0322129 (48:493)
 [13]
 , Borel preference orders in markets with a continuum of traders, J. Math. Econom. 3 (1976), 155165. MR 0439054 (55:11955)
 [14]
 , On the existence of absolutely measurable selection functions (preprint).
 [15]
 W. Yankov, Sur l'uniformisation des ensembles A, Dokl. Akad. Nauk SSSR (N.S.) 30 (1941), 597598.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197805114183
PII:
S 00029947(1978)05114183
Article copyright:
© Copyright 1978 American Mathematical Society
