Fubini Theorem Research Articles

A Fubini theorem

Let I 0 be the σ -ideal of subsets of a Polish group generated by Borel sets which have perfectly many pairwise disjoint translates. We prove that a Fubini-type theorem holds between I 0 and the σ -ideals of Haar measure zero sets and of meager sets. We use this result to give a simple proof of a generalization of a theorem of Balcerzak–Rosłanowski–Shelah stating that I 0 on 2 N strongly violates the countable chain condition.

Measures, integrals and martingales

Prelude Dependence chart Prologue 1. The pleasures of counting 2. sigma-algebras 3. Measures 4. Uniqueness of measures 5. Existance of measures 6. Measurable mappings 7. Measurable functions 8. Integration of positive functions 9. Integrals of measurable functions and null sets 10. Convergence theroems and their applications 11. The function spaces 12. Product measures and Fubini's theorem 13. Integrals with respect to image measures 14. Integrals of images and Jacobi's transformation rule 15. Uniform integrability and Vitali's convergence theorem 16. Martingales 17. Martingale convergence theorems 18. The Radon-Nikodym theorem and other applications of martingales 19. Inner product spaces 20. Hilbert space 21. Conditional expectations in L2 22. Conditional expectations in Lp 23. Orthonormal systems and their convergence behaviour Appendix A. Lim inf and lim supp Appendix B. Some facts from point-set topology Appendix C. The volume of a parallelepiped Appendix D. Non-measurable sets Appendix E. A summary of the Riemann integral Further reading Bibliography Notation index Name and subject index.

Integral conditions for nonoscillation of second order nonlinear differential equations

An extension of the Fubini theorem is presented and applied in studying the possible coexistence of nonoscillatory solutions for the differential equation ( r ( t ) | x ′ | α sgn x ′ ) ′ = q ( t ) | x | β sgn x . Some discrepancies among the cases α > β , α = β and α < β are illustrated and compared with the linear case. In addition, when α = β an application to the integral characterization of principal solutions, which answers an open problem stated in [O. Došlý, A. Elbert, Integral characterization of the principal solution of half-linear second order differential equations, Studia Sci. Math. Hungar. 36 (2000) 455–469], is given.

A Stochastic Calculus for Systems with Memory

ABSTRACT For a given stochastic process X, its segment X t at time t, represents the “slice” of each path of X over a fixed time-interval [t − r, t], where r is the length of the “memory” of the process. Segment processes are important in the study of stochastic systems with memory (stochastic functional differential equations or SFDEs). The main objective of this paper is to study nonlinear transforms of segment processes. Toward this end, we construct a stochastic integral with respect to the Brownian segment process. The difficulty in this construction is the fact that the stochastic integrator is infinite dimensional and is not a (semi)martingale. We overcome this difficulty by employing Malliavin (anticipating) calculus techniques. The segment integral is interpreted as a Skorohod integral via a stochastic Fubini theorem. We then prove Itô's formula for the segment of a continuous Skorohod-type process and embed the segment calculus in the theory of anticipating calculus. Applications of the Itô formula include the weak infinitesimal generator for the solution segment of a stochastic system with memory, the associated Feynman-Kac formula, and the Black-Scholes PDE for stock dynamics with memory.

Stochastic integral of divergence type with respect to fractional Brownian motion with Hurst parameter [formula omitted

We define a stochastic integral with respect to fractional Brownian motion B H with Hurst parameter H ∈ ( 0 , 1 2 ) that extends the divergence integral from Malliavin calculus. For this extended divergence integral we prove a Fubini theorem and establish versions of the formulas of Itô and Tanaka that hold for all H ∈ ( 0 , 1 2 ) . Then we use the extended divergence integral to show that for every H ∈ ( 1 6 , 1 2 ) and all g ∈ C 3 ( R ) , the Russo–Vallois symmetric integral ∫ a b g ( B t H ) d 0 B t H exists and is equal to G ( B b H ) − G ( B a H ) , where G ′ = g , while for H ∈ ( 0 , 1 6 ] , ∫ a b ( B t H ) 2 d 0 B t H does not exist.

A Simpler Approach to Integration and the Fubini Theorem

(2005). A Simpler Approach to Integration and the Fubini Theorem. The American Mathematical Monthly: Vol. 112, No. 1, pp. 51-60.

A Simpler Approach to Integration and the Fubini Theorem

A Simpler Approach to Integration and the Fubini Theorem

Non-unique inclusion in a flow and vast centralizer of a generic measure-preserving transformation

The problem of the inclusion in a flow is considered for a measure-preserving transformation. It is shown that if a transformation has a simple spectrum, then the set of flows including — provided that it is not empty — consists either of a unique element or of infinitely many spectrally non-equivalent flows. It is proved that, generically, inclusions in a flow are maximally non-unique in the following sense: the centralizer of a generic transformation contains a subgroup isomorphic to an infinite-dimensional torus. The corresponding proof is based on the so-called dynamical alternative, a topological analogue of Fubini's theorem, a fundamental fact from descriptive set theory about the almost openness of analytic sets, and Dougherty's lemma describing conditions ensuring that the image of a separable metric space is a second-category set.

Fonctions constructibles et intégration motivique II

We extend the formalism of an earlier Note to a global setting for which a theorem on fiber integrals and a Fubini theorem are obtained. We compare our formalism to the previous constructions given by Denef and Loeser. To cite this article: R. Cluckers, F. Loeser, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

On an extension of the Fubini theorem and its applications in ODEs

The paper deals with the change of integration for integrals∫0∞b(t)∫0tA(s)dsmdt,∫0∞A(t)∫t∞b(s)ds1/mdt,where m>0 and A,b are continuous nonnegative functions on [0,∞). There are given applications for the second-order differential nonlinear equations(a(t)|x′|αsgnx′)′+b(t)|x|βsgnx=0,where 0<β⩽α and a,b are continuous positive functions on [0,∞). The first application brings comparison results for oscillation of two generalized Emden–Fowler equations, the second one shows the similarities and discrepancies for nonoscillatory half-linear and linear differential equations.

On an extension of the Fubini theorem and its applications in ODEs

The paper deals with the change of integration for integrals ∫ 0 ∞ b(t) ∫ 0 t A(s) ds m dt, ∫ 0 ∞ A(t) ∫ t ∞ b(s) ds 1/m dt, where m>0 and A, b are continuous nonnegative functions on [0,∞). There are given applications for the second-order differential nonlinear equations (a(t)|x′| α sgn x′)′+b(t)|x| β sgn x=0, where 0< β⩽ α and a, b are continuous positive functions on [0,∞). The first application brings comparison results for oscillation of two generalized Emden–Fowler equations, the second one shows the similarities and discrepancies for nonoscillatory half-linear and linear differential equations.

The Wave Equation, Mixed Partial Derivatives, and Fubini's Theorem

In a recent paper [1] the two authors of this note have shown that Fubini's theorem on changing the order of integration and Schwarz's lemma on the equality of mixed partial derivatives are equivalent when standard assumptions of continuity and differentiability are made. The proof relies heavily upon the fundamental theorem of integral calculus as usually presented in calculus textbooks. Fubini's theorem is regarded as intuitive and easy to prove. Therefore, the equivalence established in [1] with a straightforward argument provides a simple proof of Schwarz's lemma. However, there are many cases in which the standard assumptions from which the equality of the mixed partials is derived do not hold. A typical scenario is offered by some cases of the one-dimensional wave equation.

The Wave Equation, Mixed Partial Derivatives, and Fubini's Theorem

The Wave Equation, Mixed Partial Derivatives, and Fubini's Theorem

A FUBINI THEOREM FOR GENERALIZED ANALYTIC FEYNMAN INTEGRALS AND FOURIER-FEYNMAN TRANSFORMS ON FUNCTION SPACE

In this paper we use a generalized Brownian motion process to define a generalized analytic Feynman integral. We then establish a Fubini theorem for the function space integral and generalized analytic Feynman integral of a functional F belonging to Banach algebra <TEX>$S(L^2_{a,b}[0,T])$</TEX> and we proceed to obtain several integration formulas. Finally, we use this Fubini theorem to obtain several Feynman integration formulas involving analytic generalized Fourier-Feynman transforms. These results subsume similar known results obtained by Huffman, Skoug and Storvick for the standard Wiener process.

On the associativity of the product of measure spaces

If we arbitrarily choose finitely many measure spaces, then the existence of a product of these measure spaces can be proved. It is well known that this operation is not associative which is a substantial difficulty. In this paper we show that the associative law remains true between the σ-algebras of the product spaces, and we give an extension of the Fubini's theorem.

Skew Products of Ideals

I investigate an asymmetric product construction for σ-ideals of σ-algebras of sets, corresponding to one-sided Fubini and Kuratowski-Ulam theorems, examining the cellularities of these products.

Mixed Partial Derivatives and Fubini's Theorem

(2002). Mixed Partial Derivatives and Fubini's Theorem. The College Mathematics Journal: Vol. 33, No. 2, pp. 126-130.

Mixed Partial Derivatives and Fubini's Theorem

Mixed Partial Derivatives and Fubini's Theorem

Real analysis: an introduction to the theory of real functions and integration

PART I. AN INTRODUCTION TO GENERAL TOPOLOGY SET-THEORETIC AND ALGEBRAIC PRELIMINARIES Sets and Basic Notation Functions Set Operations under Maps Relations and Well-Ordering Principle Cartesian Product Cardinality Basic Algebraic Structures ANALYSIS OF METRIC SPACES Definitions and Notations The Structure of Metric Spaces5 Convergence in Metric Spaces Continuous Mappings in Metric Spaces Complete Metric Spaces Compactness Linear and Normed Linear Spaces ELEMENTS OF POINT SET TOPOLOGY Topological Spaces Bases and Subbases for Topological Spaces Convergence of Sequences in Topological Spaces and Countability Continuity in Topological Spaces Product Topology Notes on Subspaces and Compactnes Function Spaces and Ascoli's Theorem Stone-Weierstrass Approximation Theorem Filter and Net Convergence Separation Functions on Locally Compact Spaces PART II. BASICS OF MEASURE AND INTEGRATION MEASURABLE SPACES AND MEASURABLE FUNCTIONS Systems of Sets System's Generators Measurable Functions MEASURES Set Functions Extension of Set Functions to a Measure Lebesgue and Lebesgue-Stieltjes Measures Image Measures Extended Real-Valued Measurable Functions Simple Functions ELEMENTS OF INTEGRATION Integration on C -1(W,S) Main Convergence Theorems Lebesgue and Riemann Integrals on R Integration with Respect to Image Measures Measures Generated by Integrals. Absolute Continuity. Orthogonality Product Measures of Finitely Many Measurable Spaces and Fubini's Theorem Applications of Fubini's Theorem CALCULUS IN EUCLIDEAN SPACES Differentiation Change of Variables PART III. FURTHER TOPICS IN INTEGRATION ANALYSIS IN ABSTRACT SPACES Signed and Complex Measures Absolute Continuity Singularity Lp Spaces Modes of Convergence Uniform Integrability Radon Measures on Locally Compact Hausdorff Spaces Measure Derivatives CALCULUS ON THE REAL LINE Monotone Functions Functions of Bounded Variation Absolute Continuous Functions Singular Functions BIBLIOGRAPHY

Some problems concerning interchange of integration order in functional integral formalism of U(1) gauge field theories

We show that in the functional integral formalism of U(1) gauge field theory some formal manipulation such as interchange of integration order can yield erroneous results. The example studied is analysed by Fubini theorem.