hi,

here is a little problem from Royden. There was a proposition:

Now, the problem is to show that the condition that mE1 is finite is a necessary condition, by giving a decreasing sequence <En> of measurable sets with (intersection En) = phi and mEn = inf for each n.14. Proposition: Let <En> be an infinite decreasing sequence of

measurable sets, that is, a sequence with En+1 in En for each n. Let

mE1 be finite. Then

m(intersection Ei, i :1-->inf) = lim mEn as n--> inf

I don't know... most of the decreasing sequences I found do intersect... and I found one that did not intersect in the end, but mEn is not inf...

Also, there is a little detail about writting. I've been thinking that

m(intersection Ei, i :1-->inf) is simply m( lim intersection Ei, i :1-->n as n-->inf). Is that true? because in that case, I beleive this intersection is just En since the sequence is decreasing. Then we need to find a sequence such that

m(lim En) as n-->inf = 0 while m(En) = inf for each n, which doesn't seem to be quite reasonable...

Thanks a lot!