This book presents a comprehensive study of convex analysis and measurable multifunctions, with applications in various areas of mathematics. The authors, Charles Castaing and Michel Valadier, provide a detailed exploration of convex functions, measurable multifunctions, and their properties. The text is intended for readers with a solid background in analysis, as indicated by the recommended references (Bourbaki or Dunford-Schwartz). The book covers topics such as convex lower semi-continuous functions, inf-compactness, sub-differentiability, and the Hausdorff distance. It also discusses measurable multifunctions, including selection theorems, projection theorems, and the implicit function theorem. The text includes a detailed analysis of the topological properties of the profile of measurable multifunctions with compact convex values, as well as compactness theorems for measurable selections and integral representation theorems. The book also addresses the existence of solutions to multivalued differential equations and the study of convex integrands in locally convex spaces. The authors also explore the duality of convex integral functionals in locally convex Suslin spaces and non-separable reflexive Banach spaces. The text concludes with a discussion on the natural supplement of L¹ in the dual of L and its applications, including the representation of L, singular linear functionals, and conditional expectations. The book is well-structured, with each chapter containing its own bibliography, and it serves as a valuable resource for researchers and students in functional analysis and related fields.This book presents a comprehensive study of convex analysis and measurable multifunctions, with applications in various areas of mathematics. The authors, Charles Castaing and Michel Valadier, provide a detailed exploration of convex functions, measurable multifunctions, and their properties. The text is intended for readers with a solid background in analysis, as indicated by the recommended references (Bourbaki or Dunford-Schwartz). The book covers topics such as convex lower semi-continuous functions, inf-compactness, sub-differentiability, and the Hausdorff distance. It also discusses measurable multifunctions, including selection theorems, projection theorems, and the implicit function theorem. The text includes a detailed analysis of the topological properties of the profile of measurable multifunctions with compact convex values, as well as compactness theorems for measurable selections and integral representation theorems. The book also addresses the existence of solutions to multivalued differential equations and the study of convex integrands in locally convex spaces. The authors also explore the duality of convex integral functionals in locally convex Suslin spaces and non-separable reflexive Banach spaces. The text concludes with a discussion on the natural supplement of L¹ in the dual of L and its applications, including the representation of L, singular linear functionals, and conditional expectations. The book is well-structured, with each chapter containing its own bibliography, and it serves as a valuable resource for researchers and students in functional analysis and related fields.