Mycielski-Regular Measures

Description:

Let μ be a Radon probability measure on M, the d-dimensional Real Euclidean space (where d is a positive integer), and f a measurable function. Let P be the space of sequences whose coordinates are elements in M. Then, for any point x in M, define a function ƒn on M and P that looks at the first n terms of an element of P and evaluates f at the first of those n terms that minimizes the distance to x in M. The measures for which such sequences converge in measure to f for almost every sequence are called Mycielski-regular. We show that the self-similar measure generated by a finite family of contracting similitudes and which up to a constant is the Hausdorff measure in its dimension on an invariant set C is Mycielski-regular.

Creator(s): Bass, Jeremiah Joseph
Creation Date: August 2011
Partner(s):
UNT Libraries
Collection(s):
UNT Theses and Dissertations
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Publisher Info:
Publisher Name: University of North Texas
Publisher Info: Web: www.unt.edu
Place of Publication: Denton, Texas
Date(s):
  • Creation: August 2011
Description:

Let μ be a Radon probability measure on M, the d-dimensional Real Euclidean space (where d is a positive integer), and f a measurable function. Let P be the space of sequences whose coordinates are elements in M. Then, for any point x in M, define a function ƒn on M and P that looks at the first n terms of an element of P and evaluates f at the first of those n terms that minimizes the distance to x in M. The measures for which such sequences converge in measure to f for almost every sequence are called Mycielski-regular. We show that the self-similar measure generated by a finite family of contracting similitudes and which up to a constant is the Hausdorff measure in its dimension on an invariant set C is Mycielski-regular.

Degree:
Discipline: Mathematics
Level: Doctoral
PublicationType: Doctoral Dissertation
Language(s):
Subject(s):
Keyword(s): Measure theory | probability | self-similar sets
Contributor(s):
Partner:
UNT Libraries
Collection:
UNT Theses and Dissertations
Identifier:
  • LOCAL-CONT-NO: bass_jeremiah_joseph
  • ARK: ark:/67531/metadc84171
Resource Type: Thesis or Dissertation
Format: Text
Rights:
Access: Public
Holder: Bass, Jeremiah Joseph
License: Copyright
Statement: Copyright is held by the author, unless otherwise noted. All rights reserved.