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Homework Help: Lebesgue measurable?

  1. Nov 13, 2007 #1
    1. The problem statement, all variables and given/known data
    What does it mean for a function to be lebesgue measurable?

    What does it mean for a set to be lebesgue measureable?
     
  2. jcsd
  3. Nov 13, 2007 #2

    quasar987

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    First we define the notion of an outer measure L*, which is a very intuitive generalization of the length of an interval. The outer measure of a set S in R is the inf over all collections of intervals covering S of the total length of those intervals.

    Then we define the Lebesgue measure L by restricting the domain of L* to sets that we call measurable:

    A set E is said Lebesgue measurable if for any set A, we have

    L*(A)=L*(AE)+L*(AE^c)

    This strange looking condition is a more practical characterizations due to Carathéodory of the notion of measurability introduced by Lebesgue. In either case, the condition is there to insure that the measure L will be additive. I.e. for A, B disjoint, L(AuB)=L(A)+L(B). Actually, almost all sets are measurable, and to construct one that isn't, you must make explicit use of the axiom of choice. If we work with a set theory without the axiom of choice, all sets are Lebesgue-measurable.

    And a function is measurable if the preimage of any interval is a meaurable set.
     
  4. Nov 13, 2007 #3

    matt grime

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    Surely this is defined in *any* source on measure theory. It shouldn't be the job of this forum to read the bloody book on someone's behalf.
     
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