Martingale differences are orthogonal in l2

To justify finite integral linearity independently of the affected published proof, augment every finite disjoint simple display by its zero-coefficient complement. Pairwise intersections of two augmented displays partition the whole space and carry equal coefficients wherever nonempty, so finite additivity and $0\cdot (+\mathrm{\infty })=0$ prove representation independence. Common refinements give simple monotonicity and additivity; scalar zero is handled directly and positive scalars termwise. Supremum over simple minorants, followed by increasing simple approximation and the standard sets $\{f_j\ge cs\}$ for $0<c<1$, gives monotone convergence and nonnegative additivity. Positive/negative and real/imaginary decompositions therefore give finite $L^1$ linearity. This repairs the exact foundation used by [F4] and by the cited conditional-expectation identities. For $i<j$, Cauchy–Schwarz gives $E\mathrm{\mid }{D}_i{D}_j\mathrm{\mid }\le (E{D}_i^2{)}^{1\mathrm{/}2}(E{D}_j^2{)}^{1\mathrm{/}2}<\mathrm{\infty }$. Iterating [F6] gives ${\mathcal{F}}_i\subseteq {\mathcal{F}}_{j-1}$, so [F7] makes $D_i$ measurable for the latter sigma-algebra. Both $D_j$ and $D_i{D}_j$ are integrable, so the unbounded-factor clause applies: $E[D_i{D}_j\mid {\mathcal{F}}_{j-1}]=D_{i}E[D_j\mid {\mathcal{F}}_{j-1}]=0$. Expectation preservation gives $E[D_i{D}_j]=0$. The AC assumption [F5] meets the existence hypotheses of [F2, F3, F7]. All conditional identities are identities of almost-sure classes; this argument selects no sequence of representatives.

For a fixed positive $n$, expand the finite square as ${\sum }_{k=1}^n{D}_k^2+2{\sum }_{1\le i<j\le n}{D}_i{D}_j$. Every term is integrable by the assumptions and step 1.1. Finite integral linearity makes its expectation ${\sum }_{k=1}^{n}E{D}_k^2$, because every off-diagonal term vanishes. For $n=0$ both sides are zero by the empty-sum convention, and for $n=1$ there are no off-diagonal terms.