A question about Caratheodory condition.

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In summary, the conversation discusses Theorem 16.3 from Bartle's "The Elements of Integration and Lebesgue Measure", which states the necessary and sufficient conditions for a set to be Lebesgue measurable. The question is about the condition E\subseteq A and whether it can be safely removed from the sufficient condition. The book is available online and the poster suggests posting a screenshot or scan of the relevant page. However, the issue is clarified by interpreting the statement of the theorem correctly, which assumes E \subseteq A in both directions.
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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.
 
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Perhaps you could post a screenshot or scan of the relevant page? Or at least quote the theorem?
 
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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.
 

1. What is the Caratheodory condition?

The Caratheodory condition is a mathematical concept used in the theory of measure and integration. It states that any measurable set can be approximated by a countable union of a finite collection of measurable sets. This condition is important in extending the concept of measure from simple sets to more complicated sets.

2. Why is the Caratheodory condition important?

The Caratheodory condition is important because it allows for the extension of the concept of measure to a wider range of sets, including more complicated and irregular sets. This is necessary for applications in areas such as probability, statistics, and analysis.

3. How is the Caratheodory condition used in probability theory?

The Caratheodory condition is used in probability theory to define the probability of an event as the measure of the set of all possible outcomes that satisfy that event. This allows for the calculation of probabilities for more complex events than those defined by simple sets.

4. Can the Caratheodory condition be applied to non-measurable sets?

No, the Caratheodory condition only applies to measurable sets. A set is considered measurable if it can be approximated by a countable union of intervals or other simple sets. Non-measurable sets do not have a well-defined measure.

5. Who is Caratheodory and why is this condition named after him?

Constantin Caratheodory was a Greek mathematician who made significant contributions to the fields of real analysis, measure theory, and mathematical physics. He first introduced the concept of the Caratheodory condition in his work on the theory of measure and integration. It is named after him as a tribute to his pioneering work in this area.

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