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Vitali Covering Theorem

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Definition of a Vitali Covering

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A collection of sets is called a Vitali covering of a set EE if, for every point xx in EE and every au>0 au > 0, there exists a set in the collection that contains xx and has diameter less than au au.

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Statement of the Vitali Covering Theorem for Lebesgue Measure

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Given a bounded set EE in extbfRn extbf{R}^n and a Vitali covering of EE by closed balls, there exists a disjoint subcollection of balls whose total measure differs from the measure of EE by at most au au for any given au>0 au > 0.

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Non-measurable Sets

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One application of the Vitali Covering Theorem is to show the existence of non-measurable sets, demonstrating that not all subsets of extbfR extbf{R} are Lebesgue measurable.

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Carathéodory's Criterion

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Vitali's Covering Theorem is used in the proof of Carathéodory's criterion for measurability, which characterizes the measurability of a set in terms of the outer measures of its coverings.

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Differentiation of Measures

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Vitali Covering Theorem plays a key role in the differentiation theory of measures, particularly in proving Lebesgue's Differentiation Theorem, which states that the derivative of the integral exists almost everywhere.

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