A note on the weak convergence of probability measures in the D[0,1] space

Let τ be a regular metric as defined below for the D=D[0,1] space. Even when (D,τ) is not a separable and complete metric space we show (i) that the usual conditions on a sequence of probability measures in (D,τ) ensures its weak convergence and (ii) that Prohorov's theorem in (D,τ) can be derived as a consequence of our results.