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Sets of Measure Zero

  1. Apr 5, 2012 #1
    1. The problem statement, all variables and given/known data
    Let [itex]\sigma (E)=\{(x,y):x-y\in E\}[/itex] for any [itex]E\subseteq\mathbb{R}[/itex]. If [itex]E[/itex] has measure zero, then [itex]\sigma (E)[/itex] has measure zero.




    3. The attempt at a solution
    I'm trying to show that if [itex]\sigma (E)[/itex] is not of measure zero, then there exists a point in [itex]E[/itex] such that [itex]\sigma (\{e\})[/itex] that has positive measure. But i don't know if this actually proves the question.

    I have already shown that if [itex]E[/itex] open or a [itex]G_{\delta}[/itex] set, then [itex]\sigma (E)[/itex] is also measurable. Can I use these to solve this?

    Any help is appreciated.
     
  2. jcsd
  3. Apr 5, 2012 #2

    Dick

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    The set you've defined, [itex]\sigma (E)[/itex] is a subset of R^2. [itex]\sigma (\{e\})[/itex] has zero measure. It's a line in R^2.
     
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