We are going to show that, If in , and is continuous at almost surely, then . This is based on [Patrick Billingsley 1999] ([Bil99]).
Let be the space of RCLL -valued functions, equipped with Skorohod topology. is a given probability space.
Proof: Fix . Let be a projection operator defined by . Note that, is continuous at if ([Bil99, P. 134]). Therefore, is continuous almost surely w.r.t . By mapping theorem ([Bil99, Thoerem 2.7]), implies .
Application of Proposition 1 is follows: for a continuous random function , to approximate a functional for appropriate functions and , it is enough to find in .
for all lower bounded continuous functions .
Proof: First, we assume . By Fubini’s theorem and BCT, one can write LHS as
By Proposition 1, for all . Hence, it implies and . Once again, by using Fubini’s theorem, it leads to RHS. Secondly, we assume are non-negative. Define truncated function by for all and . Then, (1) holds for all , and Monotone convergence theorem implies (1) holds for non-negative continuous . Finally, we assume satisfy for some positive constant . Then, (1) holds for , and it implies (1) holds for .