A question about Caratheodory condition.

[zzzhhh]

This question comes from Theorem 16.3 of Bartle's "The Elements of Integration and Lebesgue Measure", in page 163. The condition E\subseteq A 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, m^*(E)<+\infty can be obtained from m(A-H)=m^*(E), A-H\subseteq E can be deduced from A-E\subseteq H. Although I checked several times, I'm not sure if I missed something. So, Could someone help me make sure if the condition E\subseteq A can be safely removed from the \Leftarrow part of the theorem (then it may be the case that E\not\subseteq A 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.

 

[Landau]

Perhaps you could post a screenshot or scan of the relevant page? Or at least quote the theorem?

 

[adriank]

I looked in the book (hooray for Amazon's look inside feature). The theorem states

16.3 Theorem. Let A \subseteq \mathbf R^n be Lebesgue measurable with m(A) < +\infty. Then E \subseteq A is Lebesgue measurable if and only if m(A) = m^*(E) + m^*(A - E).​

Here m^* is the outer measure.

The problem is that you are misinterpreting the statement of the theorem. What it means is this:

Let A \subseteq \mathbf R^n be Lebesgue measurable with m(A) < +\infty, and let E \subseteq A. Then E is Lebesgue measurable if and only if m(A) = m^*(E) + m^*(A - E).​

So E \subseteq A is assumed in both directions.