Consider, for example, a shift $T$ on bi-infinite sequences $\Sigma$ on, say, two letters $\left\{a, b \right\}$. Let $\left\{p_1, p_2, \dots, p_n\right\}$, with $n\geq 3$ be a periodic $n$-cycle (i.e. $T: p_1\mapsto p_2\mapsto\cdots\mapsto p_n\mapsto p_1$). Assign the measure $\mu$ on $\Sigma$ by $\mu(\left\{p_i\right\}) = 1/n$ and for any $A\subset\Sigma$ not containing any $p_i$, $i = 1,\dots,n$, $\mu(A) = 0$. Now take two sets $A = \left\{p_1 \right\}$ and $B = \left\{p_2\right\}$. Now verify that
$ \lim_{k\rightarrow\infty} \mu(A\cap T^k(B)) = \mu(A)\mu(B) $
fails. Thus the system is not strongly mixing. Also verify that
$ \lim_{k\rightarrow\infty}\frac{1}{k}\sum_{j=0}^k\left|\mu(A\cap T^j(B)) - \mu(A)\mu(B)\right| = 0 $
also fails. Thus the system is not weakly mixing.