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In Vitali covering definition i see "derived number" word, but I dont know what that mean.

Example for vitali covering: If $f$ is strictly increasing and

$$E=\{x: \text{ there is a derived number } Df(x)<p \text{ of } f \text{ at } x\}$$

then

$$\mathcal V=\{V \in I: \lambda(f(V))<p\lambda (V) \}$$

forms a Vitali cover for E

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You’re using $V$ for two different things, and you haven’t defined $I$. I suspect that you want something like $$\mathscr{V}=\left\{V:\subseteq[0,1]:\lambda\big(f[V]\big)<p\lambda(V)\right\} \;.$$ –  Brian M. Scott Dec 20 '12 at 6:53
    
it's book note, my book don't defin I, p , Df and derived number!!! –  mshj Dec 20 '12 at 6:56
    
i supposal p is real, I is a family of nondegenerate close intervals in $setR$, and example is try to found a vitaly cover for E. λ is lebesgue measure –  mshj Dec 20 '12 at 7:01

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