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



Maury Barbato
Posts:
792
From:
University Federico II of Naples
Registered:
3/15/05


Re: C^1 Functions and Lebesgue Measure
Posted:
Jun 19, 2012 4:40 AM


> 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 28:153163), 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



