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Lebesque Measure

  1. Aug 15, 2009 #1
    Suppose that E has Lebesque measure 0. Prove that the set {x^2 : x€E} has Lebesque measure 0.

    Please help me. I have a problem which is unsolveable for me. Thanks!
  2. jcsd
  3. Aug 15, 2009 #2
    Do you know a "change of variables" formula that expresses the measure of the set {x^2 : x in E} as an integral on E ?
  4. Aug 15, 2009 #3
    Honestly i have no idea about "change of variables" formula.I try to prove with outer measure formula but i failed.In my method i have difficulties about whether x^2 is subset of E or not. I made cases for it and for x^2 subset of E i made it but i think they can be disjoint sets.That is my failure point because i have no idea about this case.
  5. Aug 15, 2009 #4
    change of variables ... you have a "nice" map [itex]\phi[/itex] that maps a set E onto a set F, and a function [itex]f[/itex] defined on F . How to relate integrals involving [tex]f[/itex] on F and [itex]f \circ \phi[/itex] on E ?
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