Consider a family of functions $\{f_n\}$, where $f_n: X \rightarrow \mathbb{R}_{\geq 0 }$, and a probability measure on $X$.
Please provide an example in which all functions $f_n$ are integrable but not uniformly integrable in the "probability sense":
$ \lim_{c \rightarrow \infty} \ \sup_n \mathbb{E} \{ f_n \mid {f_n \geq c} \} = 0$
Here there is an example, but the family is uniformly integrale in the "probability sense".