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.