Integrals Of Measurable Functions Research Articles

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.

Comment: A study of “probabilistic” and “deterministic” geostatistics

Stochastic process theory involves integrals of measurable functions over probability measure spaces. One of these is the ensemble space, Ω, whose members are sample functions on Euclidean spaceRk and the other isRk itself. What geostatisticians call the “theory of regionalized variables” is said to based on stochastic theory. A recent paper inMathematical Geology proclaims a distinction between “probabilistic” and “deterministic” geostatistics. The former is said to rely on “ensemble integrals” over Ω and the latter on “spatial integrals” overRk. This study shows that the proposed distinction rests on an arbitrary choice between two estimators for the covariance of a stochastic process; neither is an ensemble integral, both are spatial integrals, and both are Kolmogorov inconsistent. The “deterministic” estimator is identical with that of classical bivariate least-squares regression in which “spatial structure” is of no consequence. This study shows that both stochastic models are suboptimal approximations to the unique nonstationary classical statistical multivariate regression model generated by each sample pattern. The stochastic process model and its “spatial continuity measures,” thus, appear as questionable mathematical embellishments on suboptimal estimates, correspondence with geomorphic reality is tenuous, and estimates are biased and distorted. Various related misconceptions in the paper are also discussed.

Integral representation and upper bounds for stop-loss premiums under constraints given by inequalities

Abstract Upper bounds for linear functionals-especially stop-loss premiums-on sets of claim-size distributions under constraints are derived, the constraints given by equalities or inequalities for integrals of measurable functions. As an example bell-shaped distributions are considered.