Prove or disprove
In a measure space, if $\mu(X)<\infty $ and $lim_{n\rightarrow\infty}\mu (A_n)=0$ then $\mu (\bigcap_n \bigcup_{k\geq n} A_k)=0$
I have some examples supporting this but could not prove or give any counter examples.
I already know about the theorem that $\sum_n \mu (A_n)<\infty$ implies $\mu (\bigcap_n \bigcup_{k\geq n} A_k)=0$, but it did not help me.
Let $X=[0,1]$ be equipped with Lebesgue measure.
For $A_1,A_2,\cdots$ take sequence:$[0,1],[0,\frac12],[\frac12,0],[0,\frac13],[\frac13,\frac23],[\frac23,1],[0,\frac14],\dots$
Then $\lambda(A_n)\to0$ but $\bigcap_{n=1}^{\infty}\bigcup_{k=n}^{\infty}A_n=[0,1]$ (i.e. for every $x\in[0,1]$ we have $x\in A_n$ infinitely often) so has measure $1$.