WEAKLY COMPACTLY GENERATED FRECHET SPACES

It is proved that a weakly compact generated Frechet space is Lindelof in the weak topology. As a corollary it is proved that for a finite measure space every weakly measurable function into a weakly compactly generated Frechet space is weakly equivalent to a strongly measurable function.

Frechet space.Also, some consequences are obtained.All locally convex spaces are taken over the field of real numbers.By a Frechet space we mean a Haus- dorff, metrizable, complete locally convex space; we use the notations of [4] for locally convex spaces.E' will always denote the topological dual of a locally convex space E. A locally convex space is said to be weakly compactly generated if there exists an increasing sequence of G(E,E' )-compact subsets of E whose union is dense in E. THEOREM 1.Let E be a weakly compactly generated Frechet space.Then (E,G(E,E')) is a Lindelf space and E is a Borel subset of (En,G(Em,E')), E m being the bidual of E. PROOF.Let [Vn] be a sequence of O-nbd.base having the properties" (i) each V is absolutely convex and closed, n (ii) (n+l)Vn+ 1 cV n, for every n.We take [An] for an increasing sequence of weakly compact, absolutely convex subsets of E such that J A =H is dense in E. We identify (E,G(E,E')) as n=l n E' a subspace of R with product topology.R E' is a subset of the compact Haus-  ([4]).Thus f(x-hn)--+O, uniformly for f6 Vno.From this it follows that [hn] is Cauchy in E which is complete.If hn--*y in E it is easy to verify that, as elements of E x=y.This proves the claim.
Thus, in weak topology, E= (Ak+Vn) is analytic and so is Lindelf n=l k=l ([6]).Also (E,O(E,E)) can be considered as a subspace of REs.Since (Ak+Vn) is compact in R (+Vn)NE is closed in (E,o(E,ES)) and so (H+Vn) E is Borel in (E , (E , E ).Since E n=-I H+vn E'' it follows that E is Borel in (Eu,O(E,E')).
In the following result, some results and notations of ([3]) are used.Let   (X,/,) be a finite measure space, E a Hausdorff locally convex space.A , is a Baire measure on (E,((E,E')), S being the class of all Baire subsets of (E',G(E,E)) ([2], [8]).Two weakly measurable functions f.X--E, i 1,2 are said to be weakly 1 equivalent if ho fl =h f2 a.e.
[la], for every hE If E is Prechet then f" X-*E is called strongly measurable if there exists a sequence [fn] of l-simple functions, f X--*E, such that f -f, pointwise a.e. [].
n n COROLLARY 2. Let (X,/,) be a finite measure space, E a weakly corn- pactly generated Frechet space, and f :X--E a weakly measurable function.
Then f is weakly equivalent to a strongly measurable function.
REMARK.In case E is a Banach space, this result is implicit in ([2], p. 88(4), Theorem 5.4); if in addition f is bounded this is proved in ([i], p. 88).

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation