Abstract:
The main idea/ take-message from reading the thesis is to understand the change of variables formula for C1 diffeomorphisms among spaces of the same dimension. Along with this and the usage of Steiner’s symmetrization and the Isodiametric inequality, a generalization to the change of variables formula is noted, i.e, for the class of locally Lipschitz maps among spaces of different dimensions, which goes by the name of the Area and the Co-Area formula.
Description:
This document compiles some theoreMS and proofs primarily related to measure-theory and geometry in view of a part of my reading project done at TIFR CAM, in the context of my MS Thesis at IISER Pune. With the assumptions of the basic knowledge on Measure Theory, Functional Analysis, and Linear Algebra, this document proves the extension of measures from a variety of family of sets, Taylor’s formula, the Change of variables among the spaces of the same dimension, some properties of functions which are absolutely continuous and the Integration by parts. It also gives an insight via a compilation of some theoreMS and proof, of, the Fubini and Tonelli theorem( without the use of the monotone class lemma), the Radon-Nikodym theorem and the Radon Nikodym derivative, some covering theoreMS and some of the important properties of Lipschitz functions, namely the Rademacher theorem. It also has information on the dimension of fractals ( going with the name of Hausdorff dimension), the Isodiameteric inequality, which is further used to prove the change of variables formula among spaces of a different dimensions, which goes by the name of the Area and the Co-Area Formula. Most of the materials in this document are taken from the references mentioned at the end of this document.
