Show that the definition of the iterated It integral for (f in L^{2}left(mathbb{R}_{+}^{2} ight)) (Definition 20.9) is

Show that the definition of the iterated Itô integral for \(f \in L^{2}\left(\mathbb{R}_{+}^{2}\right)\) (Definition 20.9) is independent of the approximating sequence.

Data From 20.9 Definition 

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20.9 Definition (It 1951). Let f = L(R,A") and (k)ka1 CSA (R) any sequence with f. Then L(R) In(f) L(P)- lim In(ok). It is a routine exercise to show that the limit does not depend on the approximating sequence. The properties of In, which we have shown in Lemma 20.8 on 84, still hold in L(R).