WEAK REGULARITY OF PROBABILITY MEASURES

This paper examines smoothness attributes of probability measures on lattices which indicate regularity, and then discusses weaker forms of regularity; specifically, weakly regular and vaguely regular. They are obtained from commonly used outer measures, and we study them mainly for the case of M() or for those components of M(.) with added smoothness prerequisites. This is a generalization ofmany concepts presented in my earlier paper (see 1]).


INTRODUCTION
Let X be an arbitrary set and a lattice of subsets of X. A() denotes the algebra generated by and M() those finitely additive measures on A().Ma() denotes those elements of M() that are c- smooth on ; while MR() denotes those elements of M() that are -regular.To each p E M() we will associate a finitely subadditive outer measure #' on P(X), and to # E M() is associated an outer measure p".The relationships between/, #', and #" on and ' (the complementary lattice) are investigated.This leads to a consideration ofweak notions ofregularity, which can be expressed in terms of #' and/z".In this respect the normal lattices are particularly important since for such lattices regularity of/z coincides with weak regularity.We show that if # N(), those/z M() such that for L,, L, Ln, L ,/z(L) inf/z(L,,) and if is complement generated then/z is weakly regular.Combining these results gives conditions for certain measures to be regular.We adhere to standard lattice and measure terminology which will be used throughout the paper (see e.g.[2-6]) and review some of this in section two for the reader's convenience.

DEFINITIONS AND NOTATIONS
Let X be an abstract set.Let be a lattice of subsets of X.We assume throughout that and X are in .If A C X, then we will denote the complement of A by A' (i.e.A' X A).If is a lattice of subsets of X, then ' {L[L } is the complementary lattice of .
LATTICE TERMINOLOGY DEFINITION 2.1.Let be a lattice of subsets ofX.We say that: 1.
is a &lattice if it is closed under countable intersections; 6() is the lattice of countable intersections of sets of. is disjunctive if and only if x E X, L , and x L imply there exists A such that x AandAL =0. 5. /; is countably compact if and only if X J L:, Li E/;, implies there exists a finite number of i=1 the L', that cover X.
6. /; is countably paracompact if, for every sequence {L,) in/; such that L, , there exists a sequence { L, } in/; such that L, c L and L 7. /; is norma if and only if A, B /; and A f'l B imply there exists C, D /; such that ACC',BCD,,andC'ND'=.

MEASURE TERMINOLOGY
Let/; be a lattice of subsets of X. M(/;) will denote the set of finite-valued, bounded, finitely additive measures on A(/;).We may clearly assume throughout that all measures are non-negative.DEFINITION 2.2.

REGULAR PROBABILITY MEASURES
Discussion of/;-regular measures (p 6 Ma(/;)) takes place in this section.Conditions for regularity and various resulting properties are examined.

OUTER lVlEASURS
In this section we consider p M(12), and associate with it certain "outer measures"/z' and #".In general, they differ from the customary induced "outer measures"/z" and/z*.We seek to investigate the interplay ofthese outer measures on the lattice and, conversely, the effect of ; on them.DEFINITION 4.1.Let/z E M() such that/z > 0 and let E be a subset ofX. 1. #'(E) inf{#(L') E C L', L 12} is a finitely-subadditive outer measure.
1. Suppose v is an outer measure and let E be a subset of X.Then E E 8,, the set of v-measurable sets, ire(A) v(A n E) + v(A E,') for all A C X.
2. v is said to be a regular outer measure if, for A, E C X, there exists E ana () ().PROPERTY 4.8.Proofs will be omitted.
PROOF.I. Suppose v is a fiitely-subaddifive re outer mure md E 6 $.Then 2. Suppose v is a tely-badfive rel outer me md v(X)= v(E)+ v(E').Let B 6 u.Th by rel, thee egs a t F C X such that F C B d v(F) v(B).The sce Hence v(F) v(F I"1 E) + v(F Cl E').Therefore E 6 .

WEAKER NOTIONS OF REGI/LARITY
Previously we have considered some properties related to 6 Ma().We now want to consider weaker notions of regularity, and see when they might coincide with regularity; and, in general, to investigate their properties and interplay with the underlying lattice.
DEFINITION &l.Let L 6 , where is a lattice of subsets of X.
NOTATION .2. Mw() the set ofwealdy regular measures of Mv() the set of vaguely regular measures of M'()

Journal of Applied Mathematics and Decision Sciences
Special Issue on Intelligent Computational Methods for Financial Engineering

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