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A question about Caratheodory condition.

  1. Aug 17, 2010 #1
    This question comes from Theorem 16.3 of Bartle's "The Elements of Integration and Lebesgue Measure", in page 163. The condition [tex]E\subseteq A[/tex] is indeed needed in the proof of necessary condition, but I did not find its usage anywhere in the proof of sufficient condition, for example, [tex]m^*(E)<+\infty[/tex] can be obtained from [tex]m(A-H)=m^*(E)[/tex], [tex]A-H\subseteq E[/tex] can be deduced from [tex]A-E\subseteq H[/tex]. Although I checked several times, I'm not sure if I missed something. So, Could someone help me make sure if the condition [tex]E\subseteq A[/tex] can be safely removed from the [tex]\Leftarrow[/tex] part of the theorem (then it may be the case that [tex]E\not\subseteq A[/tex] albeit the part of E that lies outside of A has zero measure)? Thanks!
    This book is available online, but I cannot paste its link due to rules of this forum, you can find it at gigapedia.
    Last edited: Aug 17, 2010
  2. jcsd
  3. Aug 19, 2010 #2


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    Perhaps you could post a screenshot or scan of the relevant page? Or at least quote the theorem?
  4. Aug 19, 2010 #3
    I looked in the book (hooray for Amazon's look inside feature). The theorem states
    16.3 Theorem. Let [itex]A \subseteq \mathbf R^n[/itex] be Lebesgue measurable with [itex]m(A) < +\infty[/itex]. Then [itex]E \subseteq A[/itex] is Lebesgue measurable if and only if [itex]m(A) = m^*(E) + m^*(A - E)[/itex].​
    Here [itex]m^*[/itex] is the outer measure.

    The problem is that you are misinterpreting the statement of the theorem. What it means is this:
    Let [itex]A \subseteq \mathbf R^n[/itex] be Lebesgue measurable with [itex]m(A) < +\infty[/itex], and let [itex]E \subseteq A[/itex]. Then [itex]E[/itex] is Lebesgue measurable if and only if [itex]m(A) = m^*(E) + m^*(A - E)[/itex].​
    So [itex]E \subseteq A[/itex] is assumed in both directions.
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