Young measures on topological spaces. With applications in control theory and probability theory.

*(English)*Zbl 1067.28001
Mathematics and its Applications (Dordrecht) 571. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-1963-7/hbk). xii, 320 p. (2004).

As the authors write in the preface, the aim of the book is not to give an introduction to Young measures, but to provide a starting point for further developments of the theory. In this order of ideas, the whole matter is presented in a general setting, and the results are proved under the weakest structure assumptions.

The book opens with a brief introduction to general topology and random variables, together with the basics on integrands on product spaces. Then, narrow and weak convergence of measures on a topological space are introduced.

After the introduction of Young measures, due to the general framework, four stable topologies are defined, and the Portmanteau theorem is proved. Then, special subspaces of Young measures are introduced, and density problems are approached, together with integrable Young measures. In Chapter 3, convergence in probability of Young measures is discussed, and parametrized Dudley distances are introduced. As consequence, the fiber product lemma is proved. Parametrized Lévy-Wasserstein distances are presented as well, together with the Kantorovich-Rubinshtein theorem.

The core of the book is Chapter 4, where compactness of Young measures with respect to the stable topology is analyzed. The Topsøe criterion providing a necessary and sufficient condition is proved. Analogously, the Prohorov compactness criterion is proved as well. In order to deal with sequential compactness properties, Komlós convergence is introduced and a sequential compactness sufficient condition is established. Analogously, tightness is related to Mazur-compactness. Strong tightness is then discussed. Chapter 6 is devoted to vector-valued functions. The biting lemma, weak compactness in Banach valued \(L^1\)-spaces, and Visintin’s theorem in several infinite dimensional frameworks are established.

Applications to control theory are provided, together with some measurable selection results. Integral representation results via Young measures are established as well. In the framework of the calculus of variations, Young measures are also used to establish some weak-strong lower semicontinuity of integral functionals, and Reshetnyak-type theorems for Banach-valued measures. Some new applications of the fiber product lemma for Young measures are given to control problems and dynamic programming. Finally, in the framework of locally convex spaces, it is shown that convergence in limit theorems of probability theory actually turns out to be stable. In particular, the Rényi-mixing central limit theorem, and the stable central limit theorem for a random number of random vectors are proved.

The book provides a wide bibliography on Young measures as well.

The book opens with a brief introduction to general topology and random variables, together with the basics on integrands on product spaces. Then, narrow and weak convergence of measures on a topological space are introduced.

After the introduction of Young measures, due to the general framework, four stable topologies are defined, and the Portmanteau theorem is proved. Then, special subspaces of Young measures are introduced, and density problems are approached, together with integrable Young measures. In Chapter 3, convergence in probability of Young measures is discussed, and parametrized Dudley distances are introduced. As consequence, the fiber product lemma is proved. Parametrized Lévy-Wasserstein distances are presented as well, together with the Kantorovich-Rubinshtein theorem.

The core of the book is Chapter 4, where compactness of Young measures with respect to the stable topology is analyzed. The Topsøe criterion providing a necessary and sufficient condition is proved. Analogously, the Prohorov compactness criterion is proved as well. In order to deal with sequential compactness properties, Komlós convergence is introduced and a sequential compactness sufficient condition is established. Analogously, tightness is related to Mazur-compactness. Strong tightness is then discussed. Chapter 6 is devoted to vector-valued functions. The biting lemma, weak compactness in Banach valued \(L^1\)-spaces, and Visintin’s theorem in several infinite dimensional frameworks are established.

Applications to control theory are provided, together with some measurable selection results. Integral representation results via Young measures are established as well. In the framework of the calculus of variations, Young measures are also used to establish some weak-strong lower semicontinuity of integral functionals, and Reshetnyak-type theorems for Banach-valued measures. Some new applications of the fiber product lemma for Young measures are given to control problems and dynamic programming. Finally, in the framework of locally convex spaces, it is shown that convergence in limit theorems of probability theory actually turns out to be stable. In particular, the Rényi-mixing central limit theorem, and the stable central limit theorem for a random number of random vectors are proved.

The book provides a wide bibliography on Young measures as well.

Reviewer: Riccardo De Arcangelis (Napoli)

##### MSC:

28-02 | Research exposition (monographs, survey articles) pertaining to measure and integration |

28A33 | Spaces of measures, convergence of measures |

28C15 | Set functions and measures on topological spaces (regularity of measures, etc.) |

46G12 | Measures and integration on abstract linear spaces |

46N10 | Applications of functional analysis in optimization, convex analysis, mathematical programming, economics |

49J45 | Methods involving semicontinuity and convergence; relaxation |

60B12 | Limit theorems for vector-valued random variables (infinite-dimensional case) |