Shape Analysis, Lebesgue Integration and Absolute Continuity Connections
As shape analysis of the form presented in Srivastava and Klassens textbook Functional and Shape Data Analysis is intricately related to Lebesgue integration and absolute continuity, it is advantageous to have a good grasp of the latter two notions. Accordingly, in these notes we review basic concepts and results about Lebesgue integra- tion and absolute continuity. In particular, we review fundamen- tal results connecting them to each other and to the kind of shape analysis, or more generally, functional data analysis presented in the aforemetioned textbook, in the process shedding light on important aspects of all three notions. Many well-known results, especially most results about Lebesgue integration and some results about absolute continuity, are presented without proofs. However, a good number of results about absolute continuity and most results about functional data and shape analysis are presented with proofs. Actually, most missing proofs can be found in Roydens Real Analysis and Rudins Principles of Mathematical Analysis as it is on these textbooks and Srivastava and Klassens textbook that a good portion of these notes are based. However, if the proof of a result does not appear in the aforementioned textbooks, nor in some other known publication, or if all by itself it could be of value to the reader, an effort has been made to present it accordingly.
Shape Analysis, Lebesgue Integration and Absolute Continuity Connections, NIST Interagency/Internal Report (NISTIR), National Institute of Standards and Technology, Gaithersburg, MD, [online], https://doi.org/10.6028/NIST.IR.8217
(Accessed November 29, 2023)