Working with the epsilon-delta-criterium, a family $(X_i)_{i \in I}$ on $(\Omega,A,P)$ is uniformly integrable if

i) $sup_{i \in I}<\infty$

ii) $\forall \epsilon>0$ ex. $\delta>0$ s.t. $P(A)<\delta \Rightarrow \int_{A} |X_i|dP < \epsilon $

I easily came up with an example where i) is fulfilled, but ii) isn't ($X_n=1_{[0,\frac{n}{n}]} \cdot n$).

But I am looking for an example where ii) is fulfilled but i) isn't.

I would be really happy if someone had an answer because it would help me understand why we need (i).