Before revision: Let \Omega to be one of \X^{N+1},C[0,1],D[0,1]: a product of Polish spaces, space of continuous functions from [0,1] to a subset of R^d, and space of RCLL (right-continuous with left limits) functions from [0,1] to R^d  respectively.
After revision: Let \Omega to be one of \X^{N+1},C[0,1],D[0,1]: product of Polish spaces, space of continuous functions from [0,1] to a subset of R^d, and space of RCLL (right-continuous with left limits) functions from [0,1] to R^d respectively.