# intersection of measurable sets.

Good day, I have the next theorem: Let {${E_i}$} a collection denumerable of measurable sets such that $E_1\supset E_2\supset... \supset E_n\supset...$ and $m(E_1)$ is finite, then $m(\bigcap_{i=1}^{\infty}E_i)=lim_{n\rightarrow \infty}m(E_n)$.

How is the theorem false if $m(E_1)=\infty$, with a counterexample? I think in intervals $I_k$ in some set A such that $I_k=(a,b)$, $a,b \in \mathbb{Q}$ . Can that perform?

• every interval $I_k$ in your question @mathreda has a finite measure so you cannot create a counterexample with these $I_k$ – Marios Gretsas Aug 19 '17 at 22:12

The standard counterexample is to set $E_n=(n,\infty)$ for each $n$. Then $m(E_n)=\infty$ for all $n$, but $\bigcap_{n=1}^{\infty}E_n=\emptyset$.
• What about a counterexample where only the first $E_1$ is infinite? – mrp Aug 19 '17 at 21:58
• @mrp: The result in the question is true provided that there is some $N$ such that $m(E_n)<\infty$ for $n\geq N$. So there is no such counterexample. – carmichael561 Aug 19 '17 at 21:59