Proving Lebesque Measure of {x^2 : x€E} is 0

[Mathsos]

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!

 

[g_edgar]

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 ?

 

[Mathsos]

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.

 

[g_edgar]

change of variables ... you have a "nice" map \phi that maps a set E onto a set F, and a function f defined on F . How to relate integrals involving f[/itex] on F and f \circ \phi on E ?