By Ma J. M., Qiu R. F.

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**Extra resources for 3-Manifold Containing Separating Incompressible Surfaces of All Positive Genera**

**Example text**

Indeed, for each m ∗ -open subset W which is the finite intersection of open half spaces, namely {y ∈ E ∗ : < y, xi >< αi } W = (xi ∈ D1 αi ∈ R m ≥ 1), 1≤i≤k one has f −1 (W ) ∈ F. The Lindelöf property of E m∗ ∗ allows us to derive the same conclusion for an arbitrary m ∗ -open set, which shows that f is B(E m∗ ∗ )measurable. 1 shows that f is scalarly measurable if and ∗ )-measurable. , C) and defined on E by s(x, C) = sup{< y, x >: y ∈ C} x ∈ E. , C) is identically −∞. We consider multifunctions defined on with values in E ∗ .

When the sequence of induced probability measures ν n ◦ ( f n )−1 on R+ converges weakly to some probability measure µ on R+ , the sequence ( f n ) is uniformly integrable if and only if An f n (a) dν n (a) = R+ x d ν n ◦ ( f n )−1 (x) → R+ x dµ(x) as n → ∞. 2 The condition regarding initial endowments for the core convergence theorem of Anderson [1] is imposed only on individual consumers. In contrast, to define a perfectly competitive sequence of economies, Hildenbrand [10, Chap. 2, Section 1], used a condition on the average endowments of a vanishing sequence of coalitions with the numbers of members possibly growing to infinity.

On the top of these requirements, the first notion of convergence is nothing but the weak convergence of the joint distributions of preference relations and initial endowments. That is, we require, for every bounded and continuous function h : P × R L → R, n n −1 (z) → −1 (z) as n → ∞. P ×R L h(z) d ν ◦ (χ ) P ×R L h(z) d ν ◦ χ We then write ν n ◦ (χ n )−1 → ν ◦ χ −1 weakly as n → ∞. Although the weak convergence means, roughly, that the distribution ν ◦ χ −1 can be approximated by another distribution ν n ◦ (χ n )−1 for a sufficiently large n, its precise meaning is more restricted.