Our teacher asked to construct two counterexamples for Borel-Cantelli lemma, as follows.
(a) Construct an example with $\sum\limits_{i=1}^{\infty}\mathbb P(A_i)=\infty$ where $\mathbb P\left(\bigcap\limits_n\bigcup\limits_{k\ge n} A_k\right)=0$. I guess it would make sense to choose the unit interval $[0,1)$ for $(\Omega,\mathcal{A},\mathbb P)$ with Borel-$\sigma$-algebra and uniform distribution but I do not know how construct an example.
(b) A different example with $\mathbb P(A_i)=0$, $\sum\limits_{i=1}^{\infty}\mathbb P(A_i)=\infty$ and $\mathbb P\left(\bigcap\limits_n\bigcup\limits_{k\ge n} A_k\right)=1$
I would be very thankful if you could show me how to construct two examples.
(a) Try $A_i=[0,\frac1i]$ on the probability space $(\Omega,\mathcal{A},\mathbb P)$ you suggest.
(b) I do not understand: if $\mathbb P(A_i)=0$ for every $i$, then $\sum\limits_i\mathbb P(A_i)\ne\infty$.