I am trying to find an example of a decreasing sequence $\{E_n\}$ of Lebesgue measurable sets such that $m(\bigcap ^\infty_{n=1} E_n)\neq \lim _{n\to\infty} m(E_n)$.
I feel like whether the measure of the set is finite or infinite matters. I have no idea.... I would really appreciate it if you could help me.
Thank you
On
Let $E_n = [n,+\infty).$ Then $\bigcap_{n=1}^\infty E_n = \varnothing,$ which has measure $0,$ and $\lim\limits_{n\to\infty} m\left(\bigcap\limits_{n=1}^\infty E_n\right) = +\infty.$
Now suppose instead $E_n$ are some other sets and one of $E_n$ has finite measure. No generality is lost by assuming it's $E_1.$ Now let $$ A_n = E_n \smallsetminus \bigcap_{m=n+1}^\infty. $$ Then \begin{align} & +\infty>m(E_n) \\[8pt] = {} & m\left(A_n \cup A_{n+1} \cup A_{n+2} \cup \cdots\cup \bigcap_{m=1}^\infty E_m\right) \\[8pt] = {} & m(A_n) + m(A_2) + m(A_3) + \cdots + m\left( \bigcap_{m=1}^\infty E_m \right) \\[8pt] \to {} & \bigcap_{m=1}^\infty E_m \text{ as } n\to\infty \text{ since the series converges.} \end{align}
It is well known that measures are continuous from below/above with respect to (increasing unions)/ (decreasing intersections) of measurable sets. Formally, you can show that if $E_{n+1}\subseteq E_n$ and at least one $E_k$ has finite measure, then you'll get equality. So the only alternative is either some of your $E_n$ are not measurable, or if all of them have infinite measure.
E.g. if $m$ is Lebesgue measure, then take $E_n=(n,\infty)$, so that $E_\infty=\{\emptyset\}$.