Let E be an analytic metric space, let X be a separable metric space with a regular Borel probability measure μ and let Π: E → X be a continuous map with μ(X \ Π(E)) =0. Schwartz’s lemma states that there exists a Borel cross-section for Π defined almost everywhere (μ). The equivalence classes of these Borel cross-sections are in one-to-one correspondence with the representations of the form Γ:C<sub>b</sub>(E) → L<sup>∞</sup>(μ) with Γ(f∘Π) = f for every f ∈ C<sub>b</sub>(X). The representations are also in one-to-one correspondence with equivalence classes of the minimal measures on E.
Now let E, X, and μ be as above and let Π: E → X be an onto Borel map. There exists a Borel cross-section for Π defined almost everywhere (μ). The equivalence classes of the Borel cross-sections for Π are in one-to-one correspondence with the representations of the form Γ:B(E) → L<sup>∞</sup>(μ) with Γ(f∘Π) = f for every f in C<sub>b</sub>(X), where B(E) is the C*-algebra of the bounded Borel functions on E. The representations are also in one-to-one correspondence with equivalence classes of the minimal measures on E. / Ph. D.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/38643 |
| Date | 19 June 2006 |
| Creators | Miller, Janice E. |
| Contributors | Mathematics, Olin, Robert F., Ball, Joseph A., McCoy, Robert A., Aull, Charles E., Johnson, Lee W. |
| Publisher | Virginia Tech |
| Source Sets | Virginia Tech Theses and Dissertation |
| Language | English |
| Detected Language | English |
| Type | Dissertation, Text |
| Format | iv, 51 leaves, BTD, application/pdf, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
| Relation | OCLC# 29021824, LD5655.V856_1993.M556.pdf |
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