1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: E Measurable Implies E + y Measurable
  1. Tangent to y=e^x (Replies: 4)