I have a question about Skorokhod's representation on the real line as follows. Given a sequence of Borel probability measures \mu_n converging weakly to \mu, there exists a sequence of random variable X_n converging to X in distribution. These random variables are defined on the probability space ((0,1), Borel, Leb).

Now given a Borel probability measures \nu, can we construct a random variable Z on the same probability space such that Z is independent of {X, X_1, X_2, ...}? My guess is that it is possible, because a standard probability space seems rich enough. But I do not know a book to refer to...

Thanks!
YH