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.