Oh, very nice. I've been using the approximating process in my thesis (which is in machine learning, where the emphasis is on producing algorithms that 'work' rather than results that are asymptotically correct). A proof wasn't crucial to my thesis, but it is very useful to have in hand. Much appreciated.

I meant to add - what extra properties of $\{V^N\}$ did they use?

Unfortunately your link is behind a paywall and I don't have easy access via a university library anymore. I'm quite sure my construction doesn't work for general $V^N$ converging to $W$, and that one needs to impose extra conditions on the sequence.

John McCarthy, the inventor of LISP, has a good quote about this: “Computer chess has developed much as genetics might have if the geneticists had concentrated their efforts starting in 1910 on breeding racing Drosophila. We would have some science, but mainly we would have very fast fruit flies."

I don't think so. As far as I understand, Cameron-Martin is a special case of Girsanov. If I specify the drift, Girsanov's theorem tells me the change of measure. What I want to know is: given an equivalent change of measure, is it always expressable in the form above? In short, are changes of measure via Girsanov's theorem "the only thing you can do" to change the measure?