The constructive definition usually begins with a function f, then by the process of using Riemann sums and limits, we arrive the definition of the integral of f, ∫ b a f. On the other hand, a descriptive definition starts with a primitive F satisfying certain condition(s) such as F ′ = f and F is absolutely continuous if f is Lebesgue integrable, and F is generalized absolutely continuous if f...

Here V is the tangent to M and A is the tangent to ∂M . (By the tangent, we mean, e.g., that V (X) is the unit positively oriented pseudoscalar in the tangent algebra to M at X.) Recall the important relationship V A = N , where N is the unit outward normal to M [4, p. 319]. The relationships dV = |dV |V and dA = |dA|A define the integrals componentwise as Lebesgue integrals on M and ∂M [4, p. ...

Preface In this monograph various notions of measure and content for subsets of metric spaces are considered, and both metric and topological notions of dimension. Some familiarity with basic topics in analysis such as Riemann integrals, metric spaces, and continuous functions is assumed, as in [50, 88, 160], for instance, although a number of definitions and results that are needed are reviewe...

Our paper deals with the necessary properties in Fuzzy Measures. Therefore, it belongs to Fuzzy Mathematical Analysis. The Possibility Theory was created by Zadeh in a paper of 1978. As general purpose, to obtain a way to model flexible restrictions constructed from vague pieces of information. If we study the counterparts of expectations (from Probability Theory) in Possibility Theory, we obta...

Let ẋ = f(x) be a C autonomous differential system with k ∈ N ∪ {∞, ω} defined in an open subset Ω of R. Assume that the system ẋ = f(x) is C completely integrable, i.e. there exist n−1 functionally independent first integrals of class C with 2 ≤ r ≤ k. If the divergence of system ẋ = f(x) is non–identically zero, then any Jacobian multiplier is functionally independent of the n − 1 first integ...

It is well known that the theory of probability can be treated and developed in a consistent and uniform way using the classical theory of measure and integration. Indeed, the Russian scientist Kolmogorov identified probability with normalized classical measures and used the Lebesgue theory of integration to give a logically consistent and unifying account of probability theory. In this paper, ...