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(30 June 2006) Key: citeulike:10037012
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What you’ll find in this monograph is nothing less than a complete and rigorous study of modern functional analysis. It is intended for the student or researcher who could benefit from functional analytic methods, but who does not have an extensive background in the subject and does not plan to make a career as a functional analyst. It develops the topological structures in connection with a number of topic areas such as measure theory, convexity, and Banach lattices, as well as covering the analytic approach to Markov processes. Many of the results were previously available only in works scattered throughout the literature.
measurable structures are at the foundations of probability and statistics
If we model events as sets of states of the world, then the family of events should be closed under intersections, unions, and complements. It should also include the set of all states of the world. Such a family of sets is called an algebra of sets. If we also wish to discuss the "law of averages" which has to do with the average behavior over an infinite sequence of trials, then it is useful to add closure under countable intersections to our list of desiderata. An algebra that is closed under countable intersections is a σ-algebra. A set equipped with a σ-algebra of subsets is a measurable space and elements of this σ-algebra are called measurable sets.
The reason we do not start with a measure here is that in statistical decision theory events have their own interpretation independent of any measure, and since probability is a purely subjective notion, there is no "correct" measure that deserves special stature in defining measurability.
5 Topological vector spaces
One way to think of functional analysis is as the branch of mathematics that studies the extent to which the properties possessed by finite dimensional spaces generalize to infinite dimensional spaces.
Since there is more than one topology of interest on an infinite dimensional space, the choice of topology is a key modeling decision.
7 Convexity
main theme: maximization of linear functions over subsets of a locally convex space. In economics, linear functionals are interpreted as prices, and profit maximization and cost minimization are key concepts.
10 Charges and measures
In its abstract form a measure is a set function with additivity properties that reflect the properties of length, area, and volume. The main property is additivity. The area of two regions that do not overlap is the sum of their areas. The area of a sequence of disjoint regions is the infinite series of their areas. A probability measure is a measure that assigns measure one to the entire space.
11 Integration
In modern mathematics the process of computing areas and volumes is called integration.
20 Ergodicity
Ergodic theory can be described as the discipline that studies the long run average behavior of dynamical systems.
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