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Integral over a set of measure 0

  1. Sep 20, 2010 #1
    Is the integral over a set of measure zero always equals to zero? Can the integral be undefined?
  2. jcsd
  3. Sep 20, 2010 #2


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    It may be undefined if the function itself is peculiar with infinity as its value. For ordinary functions the integral will be 0.
    Last edited: Sep 20, 2010
  4. Sep 20, 2010 #3
    Sorry, I have the question that, if Lebesgue integration, they always define the convention [tex] \infty \cdot 0 = 0 [/tex], so, in this case, even the function takes [tex] \infty [/tex] in a set of measure 0, the integral is still 0?
  5. Sep 21, 2010 #4


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    Convention is an easy way out.
  6. Sep 21, 2010 #5


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    For Lebesgue integration, that the integral over a set of measure zero is a rather trivial theorem, following from the fact that all simple functions have integral zero -- so via (what I believe is) the usual formulation, it doesn't even need to be treated as a special case.

    Riemann integration assumes the function is real-valued, so it doesn't even apply if you are considering extended-real-number-valued functions that take on the values [itex]+\infty[/itex] or [itex]-\infty[/itex].
    Last edited: Sep 21, 2010
  7. Sep 21, 2010 #6
    But apparently to me [tex] \infty \cdot 0 = 0 [/tex] should be adopted
    Else, if [tex] f [/tex] admit [tex] \infty [/tex] on set [tex] A [/tex] of measure [tex] 0 [/tex], we may use [tex] f_{n} = n [/tex] on [tex] A [/tex] to approximate [tex] f [/tex] from below, then, the integral of [tex] f_{n} [/tex] is zero, by monotone convergence theorem, the integral of [tex] f [/tex] should be zero as well. If we do not define [tex] \infty \cdot 0 = 0 [/tex], we may get inconsistency in this case?
  8. Sep 22, 2010 #7
    But what if the f_{n} are defined over sets of measure non-zero, but that the sum of the measure of those sets converges to zero?
    Last edited: Sep 22, 2010
  9. Sep 22, 2010 #8
    Sorry that I cannot get your idea exactly. Or can you tell me explicitly what is the function [tex] f [/tex] to be integrated?
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