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E Measurable Implies E + y Measurable

  1. Mar 4, 2009 #1
    The problem statement, all variables and given/known data
    Show that if a subset E or R is measurable, then each translate E + y of E is also measurable.


    Relevant equations
    E is measurable if for all subsets A of R

    [tex]m^*(A) = m^*(A \cap E) + m^*(A \cap (\textbf R \setminus E))[/tex]

    where m* is the outer measure.

    The attempt at a solution
    The result is trivial if y = 0, so supposes y is not 0. I will denote the complement of a subset of R using '. Let F = E + y. By monotonicity of m*,

    [tex]m^*(A) \le m^*(A \cap F) + m^*(A \cap F')[/tex]

    In the other direction, I have determined the following: Since E is a subset of F' and F is a subset of E', we have that [itex]A \cap E \subseteq A \cap F'[/itex] and [itex]A \cap F \subseteq A \cap E'[/itex]. By monotonicity of m*, [itex]m^*(A \cap E) \le m^*(A \cap F')[/itex] and [itex]m^*(A \cap F) \le m^*(A \cap E')[/itex]. The former inequality is no good, but the latter one isn't. I don't know how to proceed from here. Any tips?
     
  2. jcsd
  3. Mar 5, 2009 #2
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