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

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In summary, the conversation discusses the problem of proving that the set {x^2 : x€E} has Lebesque measure 0, given that E also has Lebesque measure 0. The concept of "change of variables" is mentioned as a possible approach to solving the problem, but the person asking for help is unfamiliar with this concept. The conversation ends with a question about how to relate integrals involving a function defined on a set E to a function defined on a set F that is mapped onto E using a "nice" map.
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Mathsos
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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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  • #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 ?
 
  • #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.
 
  • #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 ?
 

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

What is Lebesque Measure?

The Lebesque Measure is a mathematical concept used to measure the size or volume of sets in higher dimensions. It is a generalization of the one-dimensional concept of length, and is used extensively in analysis and geometry.

What is {x^2 : x€E}?

{x^2 : x€E} is a set notation used to represent the set of all squared values of x, where x belongs to the set E. In other words, it is the set of all values that can be obtained by squaring any element in the set E.

Why is it important to prove that the Lebesque Measure of {x^2 : x€E} is 0?

Proving that the Lebesque Measure of {x^2 : x€E} is 0 is important because it helps us understand the properties of the set E. It also allows us to make conclusions about the size and structure of the set, which can have implications in various mathematical and scientific fields.

What is the significance of the Lebesque Measure being 0?

A Lebesque Measure of 0 means that the set in question has no volume or size in the higher dimensions. This can indicate that the set is a null set or has a fractal-like structure. It can also have implications in integration and probability theory.

How do you prove that the Lebesque Measure of {x^2 : x€E} is 0?

There are several methods for proving that the Lebesque Measure of {x^2 : x€E} is 0. One way is to use the definition of Lebesque Measure and show that the set has a lower and upper bound of 0. Another approach is to use a change of variables technique and show that the set can be transformed into a set with a Lebesque Measure of 0. Ultimately, the specific method used will depend on the properties and structure of the set E.

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