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https://hdl.handle.net/2142/23263
Description
Title
Boolean-valued probabilistic metric spaces
Author(s)
Holsey, Mickey Charles
Issue Date
1989
Doctoral Committee Chair(s)
Takeuti, Gaisi
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
In this paper the Boolean valued method is used to develop a theory closely resembling the theory of probabilistic metric spaces. In this development the complete Boolean algebra used must have the form of the quotient algebra of some atomless probability space ($\Omega,{\cal A},P$) modulo its class of null sets. In this setting the real numbers in the Boolean valued model, $V\sp{\cal B}$, correspond to random variables with domain $\Omega$, and the sets which act as binary operations on $\Re$ in the model correspond to binary operations on the class of random variables with domain $\Omega$.
The next step is to generalize the theory of metric spaces by requiring the triangle inequality be satisfied for some generalized addition $\oplus$ (i.e. some commutative, associative, nondecreasing binary operation on $\Re\sp+$ which has 0 as identity). If the generalized addition is continuous then the generalized metric space has a natural topology on it which satisfies the same general properties that metric topologies satisfy.
A Boolean valued probabilistic metric space is a triple ($\tilde S,{\cal D},{\cal G}$) which corresponds to some triple ($S,d,\oplus$) in $V\sp{\cal B}$ which acts as a generalized metric space in the nonstandard model. Thus $\tilde S$ is a subset of $V\sp{\cal B},{\cal D}$ is a function which maps ordered pairs of elements of $\tilde S$ to a.e. nonnegative random variables, and ${\cal G}$ is a binary operation on the class of a.e. nonnegative random variables. The topology on the generalized metric space in $V\sp{\cal B}$ corresponds to a topology on $\tilde S$ which is uniform.
There are a large number of probabilistic metric spaces which can be isomorphically injected into some Boolean valued probabilistic metric space. This mapping is generally not a surjection due to the richer structure of the Boolean valued probabilistic metric space. There are also probabilistic metric spaces which cannot be isomorphically injected into any Boolean valued probabilistic metric space.
This paper leads to several interesting lines of inquiry. One of these is whether all uniform topological spaces may be obtained from some generalized metric space by the Boolean valued method and another is what systems in the sciences can Boolean valued probabilistic metric spaces be used to model.
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