From weak to strong types of LE1L_E^1-convergence by the Bocce-criterion

Necessary and sufficient oscillation conditions are given for a weakly convergent sequence (resp. relatively weakly compact set) in the Bochner-Lebesgue space \l1 to be norm convergent (resp. relatively norm compact), thus extending the known results for \rl1. Similarly, necessary and sufficient oscillation conditions are given to pass from weak to limited (and also to Pettis-norm) convergence in \l1. It is shown that tightness is a necessary and sufficient condition to pass from limited to strong convergence. Other implications between several modes of convergence in \l1 are also studied

Lectures on Young Measure Theory and its Applications in Economics

A quick and very extensive introduction to the subject of Young measure is given. It is based on a particular method to transfer the classical theory of narrow convergence of probability measure into a corresponding, but richer theory of narrow convergence of transition probabilities. This method centers around a Prohorov-type extension of Komlós' theorem. Applications of this theory to existence questions in economics include optimal growth, optimal consumption, Cournot-Nash equilibrium distributions, Nash equilibria in continuum games and in games with incomplete informatio

Lectures on Young Measure Theory and its Applications in Economics

this paper we work with the following hypothesis

A unifying approach to existence of Nash equilibria

this paper the ideas of [4] will be expanded considerably, and it will be shown that a whole class of Nash equilibrium results can be obtained in this way. In itself, it is not surprising that Young measure theory should play an important role in equilibrium existence questions for game theory. Rather, it seems surprising that the narrow topology for transition probabilities had not been used before for such purposes. Indeed, if we think of a set of players T , then it is standard to let each player t 2 T choose a probability measure, say ffi (t), on the set of all actions available to him/her. Therefore, the combined effect of these choices of the players is to yield a transition probability, viz. the mapping t 7! ffi (t). Since it is evident that Nash equilibrium questions for such games can be cast into the form of some fixed point problem for the ffi 's, one is led naturally to consider the topologization of the space of all transition probabilities, for which the narrow topology turns out to be an ideal candidate. As could be expected, when the set T of players is finite or countably infinite, use of the Young measure theory adds nothing of interest, for then its topology is simply equivalent to the classical narrow topology for (products of) probability measures. It is rather when T is uncountable that the Young measure topology adds new insights to the study of Nash equilibria, and this the present paper will demonstrate. To make suitable use of the Young measure topology, a key notion of mixed externality is formulated here. For some of the equilibrium results considered such a mixed externality has a known form. For other results, phrased in terms of pure Nash equilibria, the mixed externality is both new and artificial. The basic pattern is then as follows: in..