By Shigeo Kusuoka, Toru Maruyama

The sequence is designed to compile these mathematicians who're heavily drawn to getting new not easy stimuli from monetary theories with these economists who're looking powerful mathematical instruments for his or her study. loads of fiscal difficulties should be formulated as limited optimizations and equilibration in their strategies. numerous mathematical theories were offering economists with integral machineries for those difficulties bobbing up in monetary thought. Conversely, mathematicians were inspired by means of quite a few mathematical problems raised by way of fiscal theories.

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S. j =0 Now we treat the multivalued ergodic theorem in Pettis integration. )) is integrable for all e∗ ∈ E ∗ and if any scalarly integrable selection of is Pettis-integrable. The Aumann-Pettis integral of is defined by AP- dP = {P - f dP : f ∈ S Pe } where S Pe is the set of all Pettis-integrable selections of and P - f dP is the Pettis integral of f ∈ S P e . 4). 26 C. Castaing and M. 5. Assume that Eb∗ is separable. 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, +∞[.

5 See 2) of Theorem 1 in sect. 2. 6 In this note, x i denotes the i-th coordinate of vector x. 7 The term f being C 0 -class means that f is continuous. 42 Y. Hosoya Corollary 1. The condition (A-∞) at x, (B-∞) at x, and (C-∞) at x are mutually equivalent. We should mention that if q < ∞, then both conditions (B-q) and (C-q) are not the sufficient condition of (A-(q + 1)). Debreu [4] presents a counterexample. He constructs the following g1 : ⎧ ⎨( √ y 2 , √ 1 ) (if y ≥ 0) 1+y 4 1+y 4 g1 (x, y) = ⎩(0, 1) (otherwise) and claims that there is no C 2 -class real-valued function u such that Du(x, y) = λ(x, y)g1 (x, y) for any (x, y) around (0, 0).

That is, if σ : A → R\{0} is C q -class and h : x → σ (x)g1 (x), then h satisfies the above equation if and only if g1 satisfies it. Now, suppose that σ (x) = g n1(x) for all x. Then, 1 hn (x) ≡ 1, and we can easily check that the integrability condition is equivalent to the symmetry of the Antonelli matrix of h. Hence, {g1 } satisfies condition (C-q) if and only if g1 satisfies the integrability condition. As we have seen, this property ensures the existence of u such that Du = λg1 around x. Hence, we have verified the following classical result.