Advanced Integration Theory
Springer Science & Business Media, Dec 6, 2012 - Mathematics - 876 pages
Since about 1915 integration theory has consisted of two separate branches: the abstract theory required by probabilists and the theory, preferred by analysts, that combines integration and topology. As long as the underlying topological space is reasonably nice (e.g., locally compact with countable basis) the abstract theory and the topological theory yield the same results, but for more compli cated spaces the topological theory gives stronger results than those provided by the abstract theory. The possibility of resolving this split fascinated us, and it was one of the reasons for writing this book. The unification of the abstract theory and the topological theory is achieved by using new definitions in the abstract theory. The integral in this book is de fined in such a way that it coincides in the case of Radon measures on Hausdorff spaces with the usual definition in the literature. As a consequence, our integral can differ in the classical case. Our integral, however, is more inclusive. It was defined in the book "C. Constantinescu and K. Weber (in collaboration with A.
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absolutely continuous arbitrary Banach spaces band belongs to 9t belongs to C*(X bijective Cauchy sequence choose complete vector lattice converges Corollary CP(X define Definition disjoint family f belongs family a,),er following are equivalent following assertions hold function f Given Hence implies increasing sequence infimum integral ISBN Lebesgue Lebesgue measure Let F linear LP(u mapping measure on 9t metric space néIN nonempty family nullcontinuous o-complete order relation order topology order-convergent ordered vector space orthogonal p-measurable positive measure space Proof Proposition Prove the following real number resp Riesz lattice ring of sets sequence an)nen solid vector subspace subset summable Suppose supremum surjective Take f e té I té Theorem topological space ultrafilter vector lattice vector sublattice