By Charles Castaing, Marc Lavie (auth.), Shigeo Kusuoka, Toru Maruyama (eds.)

A lot of monetary difficulties will be formulated as restricted optimizations and equilibration in their strategies. numerous mathematical theories were offering economists with essential machineries for those difficulties coming up in fiscal thought. Conversely, mathematicians were encouraged through numerous mathematical problems raised by way of fiscal theories. The sequence is designed to assemble these mathematicians who're heavily drawn to getting new not easy stimuli from fiscal theories with these economists who're looking powerful mathematical instruments for his or her research.

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So that coming back to the definition of real valued integrable superadditive process, we see that, for each x ∗ ∈ E ∗ , the sequence (δ ∗ (x ∗ , Sm ))m∈N is a superadditive integrable process in L1R ( , F , P ). 1, (Sm )m∈N is called additive. Compare with [28]. In the remainder of this paper, E is a separable Hilbert space. The following lemma is a useful and has its own interest. 1. Let E be a separable Hilbert space. Let (Xn )n∈N be a bounded sequence in L1cwk(E) ( , F , P ). s. ∀x ∈ B E . 32 C.

Let (Xn )n∈N be a bounded sequence in L1cwk(E) ( , F , P ). s. ∀x ∈ B E . 32 C. Castaing and M. Lavie lim d(x, n→∞ 1 n n Xγ (i) (ω)) = d(x, X∞ (ω)) ∀x ∈ E. s. i=1 Proof. Step 1 Let (ek )k∈N be a dense sequence in B E . For each fixed k ∈ N and n ∈ N, pick an integrable selection uk,n of Xn such that ek , uk,n (ω) = δ ∗ (ek , Xn (ω)) ∀ω ∈ . s. s. i=1 with respect to the norm of E. s . s. 2 A more general version of Koml´os theorem for B convex reflexive Banach space is available in [21]. s. s. s.

Let B be a sub-σ -algebra of F and let X be a cwk(E)-valued Pettis-integrable mapping such that E B |X| ∈ [0, +∞[. Then there exists a unique B-measurable, cwk(E)-valued Pettisintegrable multifunction, denoted by P e-E B X, which enjoys the following property: For every h ∈ L∞ (B), one has hP e-E B XdP = P e- P e- hXdP where P e- hP e-E B XdP and P e- hXdP denote the cwk(E)-valued Aumann-Pettis integral of hP e-E B X and hX respectively. 6. Assume that E and its strong dual Eb∗ are separable, T is a measurable transformation of preserving P , I is the σ -algebra of invariant sets and X : → cwk(E) is a F -measurable and Pettis-integrable mapping such that E I |X| ∈ [0, +∞[.