Lectures on Young measure theory and its applications in economics

Abstract

The rst four sections of these notes form a quick incisive introduction to the subject of Young measure theory The term Young measures refers to transition probabilities that are studied in connection with a certain weak topology ie the narrow topology for Young measures This name honors LC Young whose seminal work on generalized solutions in the calculus of variations in formed the starting point of such considerations Our presentation involves very little functional analysis and is largely based on a transfer of the classical theory of narrow convergence from the domain of probabilities section to the more general domain of transition probabilities section by means of Kconvergence and an associated key Prohorovtype extension of Komlos theorem Theorem Such an extension of Komlos theorem applies much more generally than displayed here to certain classes of abstractvalued scalarly integrable functions However in the Young measure context it is particularly eective to transfer narrow convergence properties This is because tightness a crucial condition for Theorem is under mild restrictions an automatic feature of narrow convergence of sequences of Young measures The useful portmanteau and product convergence theorems for classical narrow convergence as well as Prohorovs theorem an important device for relative narrow compactness and certain limiting support properties are thus made available for Young measures