On the probabilistic metrizability of approach spaces

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