On the probabilistic metrizability of approach spaces

We investigate approach spaces generated by probabilistic metric spaces with respect to a continuous t-norm $\ast$ on the unit interval $[0,1]$. Let $k^{\ast }$ be the supremum of the idempotent elements of $\ast$ in $[0,1)$. It is shown that if $k^{\ast }=1$ (resp. $k^{\ast }<1$), then an approach space is probabilistic metrizable with respect to $\ast$ if and only if it is probabilistic metrizable with respect to the minimum (resp. product) t-norm.