Prove both parts of Lemma 24.10 using a martingale convergence theorem (Theorem A.4). Hint: Let Fn be the σ-algebra generated by the partition Pn. Put                                       fn(x) =...

Prove both parts of Lemma 24.10 using a martingale convergence theorem (Theorem A.4). Hint: Let Fn be the σ-algebra generated by the partition Pn. Put

                                      fn(x) = E{f(X)|X ∈ An(x)}.

Then (fn, Fn) forms a convergent martingale on the probability space (Rd, Bd, µ).