Young measures on topological spaces

✍ Scribed by Castaing C., Raynaud de Fitte P., Valadier M.

• Publisher: Kluwer
• Year: 2004
• Tongue: English
• Leaves: 327
• Series: Mathematics and Its Applications
• Category: Library

✦ Synopsis

Young measures are presented in a general setting which includes finite and for the first time infinite dimensional spaces: the fields of applications of Young measures (Control Theory, Calculus of Variations, Probability Theory...) are often concerned with problems in infinite dimensional settings. The theory of Young measures is now well understood in a finite dimensional setting, but open problems remain in the infinite dimensional case. We provide several new results in the general frame, which are new even in the finite dimensional setting, such as characterizations of convergence in measure of Young measures (Chapter 3) and compactness criteria (Chapter 4). These results are established under a different form (and with fewer details and developments) in recent papers by the same authors. We also provide new applications to Visintin and Reshetnyak type theorems (Chapters 6 and 8), existence of solutions to differential inclusions (Chapter 7), dynamical programming (Chapter 8) and the Central Limit Theorem in locally convex spaces (Chapter 9).

✦ Table of Contents

Cover
......Page 1
Young Measures onTopological Spaces, With Applications in Control Theory and Probability Theory......Page 4
Contents......Page 7
Preface......Page 9
1.1 General topology......Page 12
1.2 Random elements, random sets, integrands......Page 16
1.3 Narrow and weak convergence of measures on a topological space......Page 23
1.4 Measurable cardinals and separable Borel measures......Page 28
2.1 Definitions, Portmanteau Theorem......Page 29
2.2 Special subspaces of Young measures, denseness of the space X of random variables......Page 48
2.3 Properties of ... related to the topology of T......Page 52
2.4 Integrable Young measures and Lp spaces......Page 54
3.1 Stable convergence versus convergence in probability......Page 63
3.2 Parametrized Dudley distances......Page 71
3.3 Fiber Product Lemma and applications......Page 79
3.4 Parametrized Lévy–Wasserstein distances and L1 spaces
......Page 84
4.1 Preliminary remarks and definitions......Page 93
4.2 Necessary and su.cient condition when T is separably submetrizable: Topsøe Criterion......Page 95
4.3 Flexible tightness and strict tightness: Prohorov Criterion......Page 100
4.4 Submetrizable k.–spaces, Change of Topology Lemma......Page 110
4.5 Sequential properties: sequential Prohorov property, Komlós convergence, Mazur–compactness......Page 114
4.6 Tables......Page 122
5.1 Equivalent definitions......Page 124
5.2 Strong tightness of almost everywhere convergent sequences......Page 129
6.1 Preliminaries, Biting Lemma and some basic results......Page 132
6.2 Weak convergence in L1 using Young measures......Page 150
6.3 Weak compactness and convergences in Pettis integration......Page 156
6.4 Narrow compactness of Young measures via the Dudley embedding theorem......Page 164
6.5 Support theorem for Young measures......Page 175
6.6 Visintin-type theorem in P1......Page 189
6.7 Visintin-type theorem in L1......Page 199
7.1 Measurable selection results......Page 205
7.2 Relaxed trajectories of an evolution equation governed by a maximal monotone operator......Page 208
7.3 Relaxed trajectories of a differential inclusion in a Banach space......Page 213
7.4 Integral representation theorem via Young measures......Page 217
7.5 Relaxed trajectories of a differential inclusion governed by a nonconvex sweeping process......Page 220
8.1 Weak-strong lower semicontinuity of integral functionals......Page 227
8.2 Reshetnyak–type theorems for Banach-valued measure......Page 248
8.3 Some new applications of the Fiber Product Lemma for Young measures......Page 258
9.1 Weak limit theorems in locally convex spaces......Page 279
9.2 More on stable convergence......Page 283
9.3 R´enyi–mixing Central Limit Theorem for a–mixing sequences......Page 289
9.4 Stable Central Limit Theorem for a random number of random vectors......Page 292
References......Page 302
Subject Index......Page 322
Index of Notations......Page 326