# Shape Analysis, Lebesgue Integration and Absolute Continuity Connections

## TL;DR

This paper reviews the fundamental concepts of Lebesgue integration and absolute continuity, highlighting their connections to shape analysis and functional data analysis, with some proofs included for key results.

## Contribution

It provides a comprehensive overview linking Lebesgue integration, absolute continuity, and shape analysis, including proofs of key results often omitted in other texts.

## Key findings

- Connections between Lebesgue integration and shape analysis clarified
- Key results about absolute continuity are proved and explained
- Foundational concepts are linked to practical shape analysis applications

## Abstract

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 integration and absolute continuity. In particular, we review fundamental results connecting them to each other and to the kind of shape analysis, or more generally, functional data analysis presented in the aforementioned 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 classic 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.