By Charles Castaing

ISBN-10: 4431684506

ISBN-13: 9784431684503

ISBN-10: 4431684522

ISBN-13: 9784431684527

A lot of financial difficulties should be formulated as limited optimizations and equilibration in their ideas. a variety of mathematical theories were providing economists with necessary machineries for those difficulties bobbing up in monetary thought. Conversely, mathematicians were encouraged through a variety of mathematical problems raised by way of monetary theories. The sequence is designed to compile these mathematicians who're heavily attracted to getting new difficult stimuli from monetary theories with these economists who search potent mathematical instruments for his or her researchers. The editorial board of this sequence contains the next sought after economists and mathematicians: **Managing Editors**: S. Kusuoka (Univ. Tokyo), T. Maruyama (Keio Univ.); **Editors**: R. Anderson (U.C. Berkeley), C. Castaing (Univ. Montpellier), F. H. Clarke (Univ. Lyon I), G. Debreu (U.C. Berkeley), E. Dierker (Univ. Vienna), D. Duffie (Stanford Univ.), L.C. Evans (U.C. Berkeley), T. Fujimoto (Okayama Univ.), J.-M. Grandmont (CREST-CNRS), N. Hirano (Yokohama nationwide Univ.), L. Hurwicz (Univ. of Minnesota), T. Ichiishi (Ohio country Univ.), A. Ioffe (Israel Institute of Technology), S. Iwamoto (Kyushu Univ.), okay. Kamiya (Univ. Tokyo), okay. Kawamata (Keio Univ.), N. Kikuchi (Keio Univ.), H. Matano (Univ. Tokyo), okay. Nishimura (Kyoto Univ.), M. okay. Richter (Univ. Minnesota), Y. Takahashi (Kyoto Univ.), M. Valadier (Univ. Montpellier II), M. Yano (Keio Univ).

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**Extra info for Advances in Mathematical Economics**

**Example text**

T Then KJ(T,X)::; max{l + [ f[ f J(t,ux,p,'Y(p)(t),s,z)dp,t(s)] d'1(p,Mz)] dt ILEA T JIZ Js + HJ(T + a, UX,P,'Y(IL)(T + T U an with HJ(T + a, UX,P,'Y(IL)(T + a) ::; sup ILEA 11 T+U [f [ f J(t, VX,P,'Y(IL) (t), s, z) P,t(s)] d'1(p,)t(z)]dt. Jz Js 36 C. Castaing, P. Raynaud de Fitte Here VX,/l-,"i(JL) is the trajectory on [7 + a, 1] associated with (-;:Y(JL2),JL2) with the initial condition VX,/l-,"i(/l-) (7 + a) = UX,/l-,"i(/l-) (7 + a). As above, by continuity of the integral functionals under consideration and by compactness of A, there exists JL 1 E A such that Here UX,'Y-(') 1 is the trajectory on [7,7 + a] associated with (-;:Y(JL 1), JL1) and /l- ,/lsimilarly there is JL2 E A such that 1 1 T+a [f[ f J(t,V X,/l-2,"i(/l-2)(t),S,Z)<0(/12)t(S)]<0(JL2Mz)]dt Jz Js = max /l-EA 11 T+a [f[ f J(t,vX,/l-,"i(/l-)(t),s,z)djit(s)]d,(JL)t(z)]dt.

2) It is worth to mention that the preceding result holds in the particular case where 7t is the set of all Lebesguemeasurable mappings from [0, 1] to MV§) endowed with the vague topology a(C(§)*, C(§)). In the sequel of this paper, we will make the following assumptions. Fiber product of Young measures, application to a control problem 29 (Hd H = Y([O, 1J; §) and K = Y([O, 1J; Z). In particular, the sets Hand K are decomposable and compact for the stable topology. , s, z))(s,z)E§XZ is equicontinuous; J : [0,1J x lE x § x Z -+ lR.

It is then shown in Theorem 11, p. 314, that a regular Baire measure on a compact Hausdorff space can be extended uniquely to a Borel measure and that for each Borel set E E B( K) the Borel measure J-L( E) is the supremum of the measures of the compact sets contained in E. Now a compact set in a Hausdorff space is closed and the above imply that for all A E B(K) and c ::::: 0 there is a closed set F E F such that F ~ A and J-L(A \ F) ::; c. It similarly follows that since AC E B(K) there exists G E 9 such that GC ~ AC and J-L(N \ GC) = J-L( G \ A) ::; c.

### Advances in Mathematical Economics by Charles Castaing

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