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Shape Analysis, Lebesgue Integration and Absolute Continuity Connections



Javier Bernal


As shape analysis of the form presented in Srivastava and Klassen’s 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 Royden’s “Real Analysis” and Rudin’s “Principles of Mathematical Analysis” as it is on these textbooks and Srivastava and Klassen’s 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.
NIST Interagency/Internal Report (NISTIR) - 8217
Report Number


Shape Analysis, Lebesgue Integration, Absolute Continuity, Functional Data Analysis


Bernal, J. (2018), Shape Analysis, Lebesgue Integration and Absolute Continuity Connections, NIST Interagency/Internal Report (NISTIR), National Institute of Standards and Technology, Gaithersburg, MD, [online], (Accessed June 20, 2024)


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Created July 10, 2018, Updated November 10, 2018