Contents pages
0. Introduction* 1-5
*• First class functions around a subset of a Polish space.
A. Characterizations of first class functions around 6-11
B. Spaces of continuous functions and convergence 11-15
around sets.
C. Compact families of first class functions around a 15-25
subset of a Polish space*
**• Affine and first class functions around a subset of a 26-34
convex compact.
III. Dentability and related notions in Banach spaces.
A. Some geometrical properties of subsets of Banach 35-40
B. Extensions of the above properties to operators. 40-44
IV. Operators from L [0,1] into Banach spaces.
A. The structure of the positive face of the unit ball 45-49
of Ll.
B. Equimeasurable sets and Radon-Nikodym operators. 49-51
C. Sets of small oscillation and strongly regular 51-58
D. Sets of regular oscillation and regular operators. 58-66
v* A characterization of strongly regular Banach spaces. 67-72
IV. On w*-regular sets in dual Banach spaces.
A. On the topological structure of w*-regular sets. 73-79
B. w*-regularity and the w*-Radon-Nikodym property. 80-83
C. Regular w*-compact convex sets. 83-88
VII. On regular Banach spaces and the Radon-Nikodym property.
A. On the structure of regular Banach spaces. 89-91
B. When regular sets are dentable. 91-94
C. Regularity in preduals of Von Neuman algebras. 94-96
VIII. Examples and Remarks.
A. Some topological counterexamples. 97-100
B. Some examples of operators on L . 100-107
C. Some geometrical counterexamples. 107-110
IX. Appendix: A property of preduals of Von Neuman 111-112
algebras (d'apres U. Haagerup).
X. References. 113-116
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