TITLE: Construct a sequence of measureable sets $E_1\supseteq E_2 \supseteq E_3 \supseteq \cdots$ such that $\mu(E_n)=\infty$ for each $n$ but ...
QUESTION [6 upvotes]: Construct a sequence of measureable sets $E_1\supseteq E_2 \supseteq E_3 \supseteq \cdots$ such that $\mu(E_n)=\infty$ for each $n$ but $$\mu\left(\bigcap_{n=1}^\infty E_n\right)=0$$
Claim: Let 
\begin{equation*}
\begin{aligned}
E_1= & \left(\frac{1}{i},1\right]\cup \left(\frac{1}{i+1},2\right] \cup \cdots \\
E_2= & \left(\frac{1}{i},2\right]\cup \left(\frac{1}{i+1},3\right] \cup \cdots \\
\vdots & \vdots \\
E_n= & \left(\frac{1}{i},n\right]\cup \left(\frac{1}{i+1},n+1\right] \cup \cdots \\
\vdots & \vdots \\
\end{aligned}
\end{equation*}
where $i$ is an arbitrary positive integer.
I believe that this sequence of sets satisfies the conditions above, but I want to formally write it out.

REPLY [4 votes]: The question is actually whether the example given works. Notice that $(1/i,1]\subset \bigcap_{n=1}^\infty E_n$ and so it is doesn't work.