Proving lim m(Ei) = m(E) in Measure Spaces

[jem05]

Homework Statement

Let (X , M, m) be a certain measure space and {E_n} sets in M with the property:
$$\underline{lim}$$ (E_k) = $$\overline{lim}$$(E_k) = E
prove that lim m(E_i) exists and = m(E) as n approaches infinity.

Homework Equations

The Attempt at a Solution

i solved the problem, if it were given that m($$\bigcup$$E_i) is stirctly less that infinity, but i don't know how to overcome that problem since that was not given.
any help is appreciated.
thank you.

 

[ystael]

For the finite version you've proved already, it should be enough to have the hypothesis that $$m\left(\bigcup_{j\geq n} E_j\right) < \infty$$ for some sufficiently large $$n$$, because your problem isn't changed if you throw away finitely many of the $$E_j$$ from the front. But the negation of this hypothesis is that $$m\left(\bigcup_{j\geq n} E_j\right) = \infty$$ for every $$n \in \mathbb{N}$$, which is a little stronger. Does that help?