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A Functional Analytic Proof of a Selection Lemma

Published online by Cambridge University Press:  20 November 2018

L. W. Baggett  and
Arlan Ramsay
L. W. Baggett
Affiliation:
University of Colorado, Boulder, Colorado
Arlan Ramsay
Affiliation:
University of Colorado, Boulder, Colorado
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If r is a mapping of a set X onto a set Y, then a selection for r is a mapping S of Y into X for which r o s is the identity. Analogously, if F is a mapping of a set Y into the power set 2Z of a set Z, then a selection for F is a mapping f of Y into Z such that ƒ(y) is an element of F(y) for all y in Y. These two notions are formally equivalent: Given r mapping X onto Y, define F(y) = r–l(y) and Z = X. Conversely, given F mapping Y into 2Z, define X to be the subset of Y × Z consisting of the pairs (y, z) for which z belongs to F(y), and define r on X by r(y, z) = y.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 2 , February 1980 , pp. 441 - 448
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Andenaes, P. R., Hahn-Banach extensions which are maximal on a given cone, Math. Ann. 188 (1970), 90–96.Google Scholar
2. Bourbaki, N., Topologie générale, Chapitre 9, Deuxième edition, (Hermann, Paris, 1958).Google Scholar
3. Christensen, J. P. R., Topology and borel structure (North Holland, Amsterdam, 1974).Google Scholar
4. Dixmier, J., Dual et quasi-dual d'une algèbre de Banach involutive, Trans. Amer. Math. Soc. 104 (1962), 278–283.Google Scholar
5. Effros, E. G., Convergence of closed subsets in a topological space, Proc. Amer. Math. Soc. 16 (1965), 929–931.Google Scholar
6. Effros, E. G., The Borel space of von Neumann algebras on a separable Hilbert space, Proc. J. Math. 15 (1965), 1153–1164.Google Scholar
7. Fédérer, H. and Morse, A. P., Some properties of measurable junctions, Bull. Amer. Math. Soc. 40 (1943), 270–277.Google Scholar
8. Himmelberg, C. J. and Van Vleck, F. S., Selections and implicit function theorems for multifunctions with Souslin graph, Bull. Acad. Polon. Sci. 19 (1971), 911–916.Google Scholar
9. Kuratowski, C., Topologie, Vol. I, 4th Edition (Polska Akademia Nauk, Warsaw, 1958).Google Scholar
10. Kuratowski, C. and Maitra, A., Some theorems on selectors and their applications to semicontinuous decompositions, Bull. Acad. Polon. Sci. 22 (1974), 877–881.Google Scholar
11. Mackey, G. W., Induced representations of locally compact groups I, Ann. Math., Ser. 2. 55 (1952), 101–139.Google Scholar
12. Mackey, G. W., Borel structures in groups and their duals, Trans. Amer. Math. Soc. 85 (1957), 265–311.Google Scholar
13. von Neumann, J., On rings of operators, reduction theory, Ann. Math. 30 (1949), 401–485.Google Scholar
14. Wagner, D. H., Survey of measurable selection theorems, Siam Jour. Control and Optimizatio. 15 (1977), 859–903.Google Scholar