This book has two main purposes. On the one hand, it provides a specialized and systematic development of the theory of lower fall, based on the concept of acceptance, in the spirit of Williams and Wally.
On the other hand, this theory is also extended to deal with unlimited quantities, which abound in practical applications. After Williams, we start with acceptable gambling groups.
From this, we derive the criteria of rationality --- avoid the loss and the sure interdependence --- and the methods of conclusion --- the natural extension --- to the lower limits (unconditional). We then study the different aspects of the resulting theory, including the concept of expectation (linear endings), boundaries, empty models, traditional logic of coupling, less oscillations, and monocular convergence.
We discuss n-monotonicity for the lowest incidence of falls, associated with lower fall with Choquet integration, and faith functions, random groups, possible measures, various integrals, symmetry, and representation theorems based on the theory of Bishop-De Leeuw. After that, we expanded the framework of accepted gambling groups to also consider unlimited amounts.
As before, we derive the criteria of rationality and methods of inference of the limits of vessels, and this time also allow adaptation. We apply this theory to the construction of extensions from the lowest falls from random quantities limited to a larger group of random quantities, based on ideas borrowed from the theory of integration Donford.
The first step is to extend less to the random quantities restricted to the complement of a free group (essentially random amounts). This extension is achieved by making a natural extension that can be stimulated by a rational intuitive that adding random amounts does not affect acceptance.
In another step, we round up unlimited random quantities through interrelated sequences. Basically, we define those that do not rely on the minimized subtraction limit on the rounding details.
We call these random quantities "infectious". We study the viability of change by cutting sequences, and arrive at a simple enough state.
In the case of 2-monotone, we create an integrative representation of the Choquet extension. For the general case, we prove that the extension can always be written as a Dunford envelope.
We conclude with some examples of theory.
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