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Lebesgue measure

  1. Apr 16, 2012 #1
    1. The problem statement, all variables and given/known data
    I just have a few quick questions about the definition of Lebesgue Measure of a function ( I just want some clarification on what I read in Royden)

    In Chapter 3 of Royden, An extended real valued function is defined as being Lebesgue measurable if its domain is measurable and if it satisfies one of the following:
    For each real number [itex]\alpha[/itex] the set
    1.) {x: f(x)>[itex]\alpha[/itex]} is measurable
    2.) {x: f(x)<[itex]\alpha[/itex]} is measurable
    3.) {x: f(x)≤[itex]\alpha[/itex]} is measurable
    4.) {x: f(x)≥[itex]\alpha[/itex]} is measurable

    1st Question: In a proposition before this definition in my book it says that if the domain of an extended real valued function is measurable then statements 1 through 4 are equivalent (the statements above). So does this mean that if a function is Lebesgue Measurable that its domain is measurable and that all 4 of the above statements hold? I am thinking yes since one of them must hold but they are all 4 equivalent.

    2nd Question: When we say a function is Lebesgue Measurable does this also mean that its image is measurable?



    Thank you for your time.
     
  2. jcsd
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