A sequence of measurable sets I want to find a sequence of measurable sets $A_k$. 
such that $A_k \subset [0,1]$, $\lim \lambda(A_k) =1$, but $\liminf A_k = \varnothing$.
There are some examples on function such as $\sin x \over x$ , but I can't apply on a set, $A_k$.
Please give me a simple example.

One technique you can use is a "running window". The sequence of intervals  $I_1=[0,1/2),I_2=[1/2,1],I_3=[0,1/3),I_4=[1/3,2/3),I_5=[2/3, 1],I_6=[0,1/4),\ldots$ has the property that their measure goes to $0$ but every element of $[0,1]$ is in infinitely many of these intervals. So you can simply let $A_k=[0,1]\backslash I_k$.
