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Homework Help: Integral using Lebesgue Measure

  1. Jun 26, 2010 #1
    1. The problem statement, all variables and given/known data
    Find the integral of the function f(x)=3 when x is rational and 2 when x is irrational on the interval [0,1].

    2. Relevant equations



    3. The attempt at a solution
    So I partition [0,1] into two disjoint sets A and B. [tex]A=[0,1] \cap Q[/tex] and [tex]B=[0,1] \cap Q^{c}[/tex].
    Now the integral should be equal to 3 length(A) + 2 length(B). Since A is a countable set then its measure is zero. But I do not understand how to calculate the measure of B since it is uncountable.
     
  2. jcsd
  3. Jun 26, 2010 #2
    But you know what the measure of [0,1] is...
     
  4. Jun 26, 2010 #3
    Thanks for the hint. So 1=len([0,1])=len(A)+len(B)=len(B).
     
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