Vitali-Besicovitch-Morse's Covering Theorems and Differentiation
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In this book will be presented (with complete proofs) 3 basic covering Theorems as the Vitali covering Theorem, the Besicovitch covering Theorem and Morse covering Theorems and the Morse covering Theorems (for covers of special subset such as balls, convex subsets, Morse subsets in the finite dimensional R^n) and applications to giving proofs of Besicovitch "derivative" theorem about the Radon-Nikodym derivative measures and the Lebesgue differentiation theorem about Lebesgue points. All the above Theorems are now considered as ones of basic technical tools for proving many fundamental theorem...
In this book will be presented (with complete proofs) 3 basic covering Theorems as the Vitali covering Theorem, the Besicovitch covering Theorem and Morse covering Theorems and the Morse covering Theorems (for covers of special subset such as balls, convex subsets, Morse subsets in the finite dimensional R^n) and applications to giving proofs of Besicovitch "derivative" theorem about the Radon-Nikodym derivative measures and the Lebesgue differentiation theorem about Lebesgue points. All the above Theorems are now considered as ones of basic technical tools for proving many fundamental theorems of modern methods in the calculus of variation for variational integral energy functional on Sobolev spaces and Orlecz-Sobolev spaces of weakly differentiable functions.
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