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Lebesgue Integral: Practice Problem

  1. Oct 14, 2014 #1
    1. The problem statement, all variables and given/known data
    Suppose that f is a nonnegative Lebesgue measurable function and E is a measurable set.
    Let A = {x ∈ E : f(x) = ∞}. Show that if ##\int_E f dλ < ∞## then ##λ(A) = 0##

    2. Relevant equations


    3. The attempt at a solution

    Let ##\phi(x)=\sum_{x\in A}a_i\mathcal{X}_{A_i}(x)## so by the construction of ##A##, ##a_i=\infty## for all ##i## and ##A=\cup A_i##
    Then
    ##\infty\ge\int_E f dλ= \int_{E-A}f dλ+\int_Af dλ=\int_{E-A}f dλ+\int \phi(x)=\int_{E-A}f dλ+\sum a_i \lambda(A_i)##
    ##\implies \lambda(A_i)=0 ## for all i
    ##\implies \lambda(A)=0##
    This seems simple and intuitive so I'm worried I'm missing something.
     
  2. jcsd
  3. Oct 14, 2014 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    No, that's pretty much the idea.
     
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