Re: Measure Theory Questions

This (what I wrote originally), sadly, is exactly how the problem was written.
I find it strange because there is a theorem that indicates something similar.
..

One major difference is that the E_k's in the theorem are such that E_k
decreases to E (in the homework it's subsets, but not such that E_k=E_k+1.
(which I took to be possible when saying "decreases to."
That is the theorem states, if you have that the sequence {E_k} of measurable
sets and E_k decreases to E, and the m(E)<infinity, then limit k->infinity m
(E_k) = m(E).

So I finally took the problem to mean construct a set such that the criteria
are met - I was thinking something Cantor like... and someone suggested an
example to me but it doesn't meet the m(E_k)<infinity part.

So, I thought that the example, while perhaps trivial, fit the problem...
does that make any sense (or is it more confusing)? I was thinking it might
have to be a Cantor like set, but that didn't seem to fit the inequality, so
I was at a bit of a loss.


3. Prove that there exist a sequence E_1, E_2, E_3, ... of sets in Reals such
that E_1 is in E_2 is in E_3 ...; E = the intersection of E_k, k = 1 ->
infinity; the outer measure m(E_k) < infinity; and lim k-> infinity m(E_k) >
m(E). Note strict inequality.


Joe Blow wrote:
No, I think what Henry was saying is that you probably wrote the question down incorrectly. As it is, it is completely trivial.

.