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A part of the proof is given below: My questions are: 1- I do not understand why "Since $f'(x) = 0$, we can find $y_{x} > o$ such that we have both $[x,y_{x}] \subseteq (a,b)$ " could anyone explain this step for me please? 2- Why the set $\mathcal{B}$ is a vitali cover of $E$? where is ...

In mathematics, the Vitali covering lemma is a combinatorial and geometric result commonly used in measure theory of Euclidean spaces. This lemma is an intermediate step, of independent interest, in the proof of the Vitali covering theorem. The covering theorem is credited to the Italian mathematician Giuseppe Vitali. The theorem states that it is possible to cover, up to a Lebesgue-negligible set, a given subset E  of Rd by a disjoint family extracted from a Vitali covering of E. == Vitali covering lemma == === Statement of the lemma === Finite version: Let B...

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