An Introduction to Measure and Integration

Rana, Inder

doi:10.1090/gsm/045

Cited by 29 publications

(11 citation statements)

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“…R n is an injection, i.e., a bijection between D and . i ; h/.D/, and assuming sufficient differentiability and rank of the corresponding Jacobian matrix for i (h by its regularity already has this property), and then using the standard transformation of probability/change-of-variable/substitution theorem ( [30,Chapter VIII], [51,Section 9.3], [20,Section 8.5], [22,Section IV.5]) to obtain the joint pdf of . i .X /; h.X// at any possible outcome .w i ; t / in .…”

Section: Overview Of the Results Of The Entire Four-part Papermentioning

confidence: 99%

Use of the global implicit function theorem to induce singular conditional distributions on surfaces in n dimensions: Part I

Bamber¹,

Goodman

Gupta³

et al. 2010

Random Operators and Stochastic Equations

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This paper stems from previous work of certain of the authors, where the issue of inducing distributions on lower-dimensional spaces arose as a natural outgrowth of the main goal: the estimation of conditional probabilities, given other partially specified conditional probabilities as a premise set in a probability logic framework. This paper is concerned with the following problem. Let 1 Ä m < n be fixed positive integers, ; ¤ D Â R n some open domain, and hW D ! R m a function yielding a full partitioning of D into a family, denoted M.h/, of lower-dimensional surfaces/manifolds via inverseh/º noting each h 1 .t / can also be considered the solution set of all X in D of the simultaneous equations h.X/ D t . Let X be a random vector (rv) over D having a probability density function (pdf) f . Then, if we add sufficient smoothness conditions concerning the behavior of h (continuous differentiability, full rank Jacobian matrix dh.X /=dX over D, etc.), can an explicit elementary approach be found for inducing from the full absolutely continuous distribution of X over D a necessarily singular distribution for X restricted to be over M.h/ that satisfies a list of natural desirable properties? More generally, for a fixed positive integer r, we can pose a similar question concerning the rv .X /, when W D ! R r is some bounded a.e. continuous function, not necessarily admitting a pdf. Problem considered hereOriginally, over thirty years ago, Higgins and Saw in a series of papers considered special lower-dimensional cases of the problem considered in the above abstract and came to the conclusion, in effect, but without formally recognizing it, that the Implicit Function Theorem (IFT) could be used to address this problem. In this paper, using a stronger global form of IFT (GIFT), it is shown that a succinct theory of inducing distributions on families, denoted M.h/ in the abstract, of lowerdimensional surfaces can be developed, extending the ideas of Higgins and Saw, using just the tools of elementary analysis, yet the proposed approach possesses a number of natural properties. More specifically, under the GIFT assumption for h, we take as the definition of E. .X / j X in h 1 .t//, for each t in range.h/, the limit of the standard conditional expectation with non-zero antecedent probability, E. .X/ j X in h 1 .a//, for any sequence or net of open neighborhoods of t in range(h) with positive probability nesting down to singleton ¹tº in a way equivalent to a nested family of open m-spheres. It is shown that not only all of Higgins and Saw's results hold concerning compatibility and connections with the rv's .X/ and h.x/, but also that: the definition obeys the Radon-Nikodym constraint in many various ways, it has a strong invariance property, satisfies an optimal L 2approximation criterion, as propounded, e.g., by Tjur, has natural connections with surface measures and surface densities, and in particular for h affine and the rv X distributed as multivariate gaussian, is completely compatible with the standard theory of sing...

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Section: Overview Of the Results Of The Entire Four-part Papermentioning

confidence: 99%

Use of the global implicit function theorem to induce singular conditional distributions on surfaces in n dimensions: Part I

Bamber¹,

Goodman

Gupta³

et al. 2010

Random Operators and Stochastic Equations

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“…It is possible to use a similar approach (by applying the sufficient conditions) in other FIS-based assessment models. In addition, it is worthwhile to investigate other properties of the length function, e.g., sub-additivity [12], for FIS-based assessment and decision making models in future work.…”

Section: Discussionmentioning

confidence: 99%

“…The importance of the monotonicity property in assessment and decision making problems, e.g., the assessment of sustainable development and measurement of material recyclability, has been described as the natural requirement in [9]. It is also possible to explain the importance of the monotonicity property from the theoretical aspect of the length function in the field of measure theory [12]. A valid comparison and/or ranking (which eventually leads to decision making) scheme among different objects/ situations based on the predicted measuring index is important [2,13].…”

Section: Introductionmentioning

confidence: 99%

On the Use of Fuzzy Inference Systems for Assessment and Decision Making Problems

Tay

Lim

2010

Handbook on Decision Making

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Abstract. The Fuzzy Inference System (FIS) is a popular paradigm for undertaking assessment/measurement and decision problems. In practical applications, it is important to ensure the monotonicity property between the attributes (inputs) and the measuring index (output) of an FIS-based assessment/measurement model. In this chapter, the sufficient conditions for an FIS-based model to satisfy the monotonicity property are first investigated. Then, an FIS-based Risk Priority Number (RPN) model for Failure Mode and Effect Analysis (FMEA) is examined. Specifically, an FMEA framework with a monotonicity-preserving FIS-based RPN model that fulfils the sufficient conditions is proposed. A case study pertaining to the use of the proposed FMEA framework in the semiconductor industry is presented. The results obtained are discussed and analyzed.

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“…As promised in the above proof, let us state the following measure theoretic lemma, whose proof uses the classical Vitali convergence theorem [30,Theorem 8.5.14], and is exactly the same as the argument of [15,Lemma 5.3]:…”

Section: Proof Of Theorem 12 and Theorem 11mentioning

confidence: 99%

Comparison of the Calabi and Mabuchi geometries and applications to geometric flows

Darvas¹

2017

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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An Introduction to Measure and Integration

Cited by 29 publications

References 0 publications

Use of the global implicit function theorem to induce singular conditional distributions on surfaces in n dimensions: Part I

Use of the global implicit function theorem to induce singular conditional distributions on surfaces in n dimensions: Part I

On the Use of Fuzzy Inference Systems for Assessment and Decision Making Problems

Comparison of the Calabi and Mabuchi geometries and applications to geometric flows

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