This item is only available for download by members of the University of Illinois community. Students, faculty, and staff at the U of I may log in with your NetID and password to view the item. If you are trying to access an Illinois-restricted dissertation or thesis, you can request a copy through your library's Inter-Library Loan office or purchase a copy directly from ProQuest.
Permalink
https://hdl.handle.net/2142/20628
Description
Title
Hyperfinite transversal theory
Author(s)
Zivaljevic, Bosko T.
Issue Date
1989
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
A famous theorem of P. Hall gives a necessary and sufficient condition for a bipartite graph $\Gamma$ to possess a matching f. (A bipartite graph $\Gamma$ is a subset of the cartesian product of two finite sets X and Y. A matching f of $\Gamma$ is a 1-1 function f which is a subset of $\Gamma$ and which has the same domain as $\Gamma$.) The graph $\Gamma$ can be thought of as a finite family of finite sets $\{\Gamma(x) : {x}{\in}{X})\}$. Thus, the existence of a matching of $\Gamma$ is equivalent to the existence of a system of distinct representatives of the corresponding family of finite sets, i.e., to the existence of an injective choice function of the family $\{\Gamma(x) : {x}{\in}{X})\}$. The part of Graph Theory which deals with refinements and variants of P. Hall's theorem is called Transversal Theory.
"In our setting Hall's type result are proved in the case when the sets X and Y are considered to be Loeb measurable sets with respect to some uniformly distributed hyperfinite counting measure. Hall's condition is replaced by its measure theoretic analogue and for the graph $\Gamma$ we use sets of different ""complexity"" (in the sense of Descriptive Set Theory). At the same time the choice function, i.e., the matching in the statement of Hall's theorem, is required to be of ""nice"" structure, that is we ask that the function be internal. The terminology is likewise adjusted to the measure case situation. Accordingly, a matching need not have its domain equal to the domain of the graph, as was asked in the conclusion of the discrete case of Hall's theorem, but, rather, the domain of the matching should be of full measure in the domain of the graph."
Use this login method if you
don't
have an
@illinois.edu
email address.
(Oops, I do have one)
IDEALS migrated to a new platform on June 23, 2022. If you created
your account prior to this date, you will have to reset your password
using the forgot-password link below.
