Second moment is the expected predictable quadratic variation

By [F1], $Z_n=M_n^2-⟨M{⟩}_n$ is integrable and $E[Z_k\mid {\mathcal{F}}_{k-1}]=Z_{k-1}$ for each $k\ge 1$. Expectation preservation gives $E{Z}_k=E{Z}_{k-1}$. Induction over the finitely many times up to $n$ yields $E{Z}_n=E{Z}_0=E{M}_0^2$, also at $n=0$. The invocations of F1 and F2 are made under the AC assumption F4.

For the finite integral linearity used here, augment each finite disjoint display of a nonnegative simple function by its zero-coefficient complement. Pairwise intersections of two augmented displays partition the whole space and have equal coefficients on nonempty cells, so finite additivity and $0\cdot (+\mathrm{\infty })=0$ prove representation independence. Common refinements give simple addition and monotonicity, while scalar zero is direct and positive scalars are termwise. Supremum over simple minorants, increasing simple approximation and the sets $\{f_j\ge cs\}$ for $0<c<1$ give monotone convergence and nonnegative additivity. Positive/negative and real/imaginary decompositions then give finite $L^1$ linearity. The square is integrable by the square-integrability hypothesis, and the bracket is integrable by [F5], so this local linearity gives $E{Z}_n=E{M}_n^2-E⟨M{⟩}_n$. Rearranging the finite equality from step 1.1 proves the formula. If $M_0=0$ its second moment is zero. At $n=0$ the bracket is zero and the equation reads $E{M}_0^2=E{M}_0^2$.