The martingales difference

Question

I have a question about martingales. $(X_n,F_n)$ is a martingale. $$D_n=X_n-X_{n-1}$$ Proof that each pair of $D_n$ are uncorrelated.

I have a solution that for $m<n$ we have: $$ED_{m}D_{n}=E(E(D_{m}D_{n}|F_{n-1}))=E(D_mE(D_{n}|F_{n-1}))=0$$

I don't understand this part: $$E(E(D_{m}D_{n}|F_{n-1}))=E(D_mE(D_{n}|F_{n-1}))$$ $D_m$ is obviously $F_n$ measurable. $D_m$ is limited? Because only then we can write such a equation.

Thanks in advance.

Answer 1

You actually do not need a uniform bound on $D_m$: all the written expectations are finite if for each $m$, $D_m^2$ has a finite expectation.