On weak convergence implying strong convergence under extremal conditions. (English) Zbl 0768.46013

T. Rzeżuchowski [Bull. Aust. Math. Soc. 39, No. 2, 201-214 (1989; Zbl 0651.28007)] showed that under a certain extremal subset condition the weak convergence in an \(L^ 1\)-space does imply the strong one. The author has improved upon and extended Rzeżuchowski’s result to the infinite-dimensional case by utilizing the theory of Young measures. His theorem also yields the main result of [same author, ibid. 33, 363-368 (1986; Zbl 0579.46018)].
Reviewer: T.Kubiak (Poznań)


46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B22 Radon-Nikodým, Kreĭn-Milman and related properties
Full Text: DOI


[1] Balder, E. J., A general approach to lower semicontinuity and lower closure in optimal control theory, SIAM J. Control Optim., 22, 570-598 (1984) · Zbl 0549.49005
[2] Balder, E. J., On weak convergence implying strong convergence in \(L_1\)-spaces, Bull. Austral. Math. Soc., 33, 363-368 (1986) · Zbl 0579.46018
[3] Balder, E. J., Generalized equilibrium results for games with incomplete information, Math. Oper. Res., 13, 265-276 (1988) · Zbl 0658.90104
[4] Balder, E. J., On Prohorov’s theorem for transition probabilities, Sém. Anal. Convexe, 19, 9.1-9.11 (1989) · Zbl 0732.60007
[5] Balder, E. J., On equivalence of strong and weak convergence in \(L_1\)-spaces under extreme point conditions, (Preprint, No. 599. Preprint, No. 599, Israel J. Math. (1989), Mathematical Institute, University of Utrecht), in press · Zbl 0758.28005
[6] Berliocchi, H.; Lasry, J. M., Intégrandes normales et mesures paramétrées en calcul des variations, Bull. Soc. Math. France, 101, 129-184 (1973) · Zbl 0282.49041
[7] Brooks, J. K.; Dinculeanu, N., Weak compactness in spaces of Bochner integrable functions and applications, Adv. in Math., 24, 172-188 (1977) · Zbl 0354.46026
[8] Castaing, C., Convergence faible et sections extrémales, Sém. Anal. Convexe, 18, 2.1-2.18 (1988) · Zbl 0698.46024
[9] Castaing, C.; Valadier, M., Convex analysis and measurable multifunctions, (Lecture Notes in Mathematics, Vol. 580 (1977), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0346.46038
[10] Diestel, J.; Uhl, J., Vector Measures, (Math. Surveys, Vol. 15 (1977), Amer. Math. Soc: Amer. Math. Soc Providence, RI) · Zbl 0369.46039
[11] Floret, K., Weakly compact sets, (Lecture Notes in Mathematics, Vol. 801 (1980), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0232.46004
[12] Ionescu-Tulcea, A.; Ionescu-Tulcea, C., Topics in the Theory of Lifting (1969), Springer-Verlag: Springer-Verlag Berlin · Zbl 0179.46303
[13] Laurent, J.-P, Approximation et Optimisations (1972), Hermann: Hermann Paris · Zbl 0238.90058
[14] Neveu, J., Mathematical Foundations of the Calculus of Probability (1965), Holden-Day: Holden-Day San Francisco · Zbl 0137.11301
[15] Rzez̆uchowski, T., Strong convergence of selections implied by weak, Bull. Austral. Math. Soc., 39, 201-214 (1989) · Zbl 0651.28007
[16] Rzez̆uchowski, T., Impact of dentability on weak convergence (1990), Institute of Mathematics, Warsaw Technical University, Preprint
[17] Schwartz, L., Radon Measures (1973), Oxford Univ. Press: Oxford Univ. Press London
[18] Tartar, L., Compensated compactness and applications to partial differential equations, (Knops, R. J., Nonlinear Analysis and Mechanics: Heriot-Watt Symposium. Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Research Notes in Mathematics, Vol. 39 (1979), Pitman: Pitman London), 136-212 · Zbl 0437.35004
[19] M. Valadier; M. Valadier · Zbl 0738.28004
[20] Valadier, M., Différents cas oú, grace á une propriété d’extremalité, une suite de fonctions intégrables faiblement convergente converge fortement, Sém. Anal. Convexe, 19, 5.1-5.2 (1989) · Zbl 0734.49004
[21] Visintin, A., Strong convergence results related to strict convexity, Comm. Partial Differential Equations, 9, 439-466 (1984) · Zbl 0545.49019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.