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Basic (Lebesgue) integration

  1. May 18, 2009 #1
    1. The problem statement, all variables and given/known data

    Let (X,m) be a measure space, f:X->[0, infinity] be measurable.

    If the integral over X of f is 0, show that f=0 almost everywhere



    2. Relevant equations



    3. The attempt at a solution

    Suppose that f is nonzero on A, m(A)>0.

    I've reduced problem to proving that the following is impossible:

    f is positive on a set of nonzero measure and
    {x such that f(x)>epsilon} has zero measure for all epsilon greater than zero.

    I don't know how to do that though...

    Any help?
     
  2. jcsd
  3. May 18, 2009 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Suppose the statement is not true- that there exist a set A, of measure greater than 0, on which f is positive. Then the integral of f over A alone is positive and, since f is never negative, the integral of f over X is not less than the integral of f over A.
     
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