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

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Homework Help Overview

The discussion revolves around proving that the set {x^2 : x ∈ E} has Lebesgue measure 0, given that the set E has Lebesgue measure 0. The subject area involves measure theory and properties of Lebesgue measure.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the application of a "change of variables" formula to relate the measure of the set {x^2 : x ∈ E} to an integral on E. Some express uncertainty about the formula and its implications. Others mention difficulties with the outer measure formula and the relationship between the sets involved, particularly questioning whether x^2 is a subset of E.

Discussion Status

The discussion is ongoing, with participants exploring different approaches and expressing confusion about certain concepts. There is no explicit consensus, but some guidance regarding the change of variables and its application has been suggested.

Contextual Notes

Participants are grappling with the definitions and properties of Lebesgue measure, as well as the implications of the change of variables in the context of measure theory. There are indications of missing information regarding the nature of the sets involved and the assumptions being made.

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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!
 
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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 ?
 
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.
 
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 ?[/tex]
 

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