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Actually calculating the Lebesgue Outer Measure of a set

  1. Mar 17, 2012 #1
    mtheory-1.png

    If you are unfamiliar with that notation, S(A, B) = (A \ B) U (B \ A), which is the symmetric difference.

    And D(A, B) = m^*(S(A, B)), which is the outer measure of the symmetric difference.

    My issue in this calculation is this: outer measure of a set A is defined in terms of a countable covering of A with intervals I_k. How do you cover something in R^2 with intervals?

    My guess on how to solve the problem: D(A, B) = m^*(S(A, B)) = m^*((0, 1]X[0, 1] U [1, 4]X(1, 10) U [0, 1]X[2, 10]) = m^*(0, 1]X[0, 1] + m^*[1, 4]X(1, 10) + m^*[0, 1]X[2, 10] = 1 + 27 + 8 = 36
     
  2. jcsd
  3. Mar 18, 2012 #2

    jgens

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    The outer Lebesgue measure on [itex]\mathbb{R}[/itex] can be defined in terms of a countable covering by intervals. The outer Lebesgue measure on [itex]\mathbb{R}^2[/itex] can be defined in terms of a countable covering by rectangles. This should clear up your confusion.

    If you want more information about this, you can read the following wikipedia article: http://en.wikipedia.org/wiki/Lebesgue_measure#Construction_of_the_Lebesgue_measure
     
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