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Topic: C^1 Functions and Lebesgue Measure
Replies: 1   Last Post: Jun 19, 2012 4:40 AM

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Maury Barbato

Posts: 791
From: University Federico II of Naples
Registered: 3/15/05
Re: C^1 Functions and Lebesgue Measure
Posted: Jun 19, 2012 4:40 AM
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> Hello,
> let f:R^n -> R^m be a C^1 function such that the
> jacobian
> J of f in every point x has rank m, and let E be a
> subset
> of R^m with Lebesgue measure zero.
> Is it true that f^{-1}(E) has Lebesgue measure zero
> in
> R^n?
> Thank you very very much for your attention.
> My Best Regards,
> Maury Barbato

I found an answer to my question
in "Submersions and preimages of sets of measure zero" (Siberian Mathematical Journal 28:153-163), S.P. Ponomarev proves as Theorem 1 that if ??R^m is open and g?C^r (?,R^n ) where r ? m?n+1 then g^{?1} preserves Lebesgue measure 0 sets if and only if g is almost everywhere a submersion (i.e., the Jacobian derivative g ? has rank n almost everywhere).

Thank you very much for your attention.
My Best Regards,
Maury Barbato

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