Martingales and martingale differences correspond

First reconstruct the finite-integral interface used below. Augment every finite disjoint display of a nonnegative simple function by the complement of its displayed sets with coefficient $0$. Intersections of two augmented displays partition the whole space, and equality of the functions makes their coefficients agree on every nonempty cell. Finite additivity and $0\cdot (+\mathrm{\infty })=0$ therefore prove representation independence. Common augmented refinements give monotonicity and additivity term by term; homogeneity is direct when the scalar is zero and termwise when it is positive. Taking suprema over simple minorants gives nonnegative monotonicity, and increasing simple approximations together with the sets $\{f_j\ge cs\}$, $0<c<1$, give monotone convergence. Applying this to sums of increasing simple approximants gives nonnegative additivity. Positive/negative and real/imaginary decompositions now give the finite real and complex $L^1$ linearity used in [F3]. With this replacement for the affected foundation, the event-integral construction and uniqueness proof of the cited conditional-expectation algebra apply. For a martingale $M$, both $M_n$ and $M_{n-1}$ are ${\mathcal{F}}_n$-measurable. Their difference $D_n$ is measurable and integrable, with $E\mathrm{\mid }{D}_n\mathrm{\mid }\le E\mathrm{\mid }{M}_n\mathrm{\mid }+E\mathrm{\mid }{M}_{n-1}\mathrm{\mid }<\mathrm{\infty }$. By linearity and conditioning the known $M_{n-1}$, $E[D_n\mid {\mathcal{F}}_{n-1}]=M_{n-1}-M_{n-1}=0$. This meets the difference definition.

Conversely every summand $D_k$ for $k\le n$ is ${\mathcal{F}}_n$-measurable, as is $M_0$. The finite sum is adapted and integrable by [F3]–[F4]. Its next increment is $D_{n+1}$, so $E[M_{n+1}\mid {\mathcal{F}}_n]=E[M_n+D_{n+1}\mid {\mathcal{F}}_n]=M_n+0=M_n$. Thus it is a martingale Martingale submartingale and supermartingale. AC is inherited from the conditional classes; if versions of a class sequence are to be chosen, AC permits those countably many selections.

The finite identities ${\sum }_{k=1}^n(M_k-M_{k-1})=M_n-M_0$ and $(M_0+{\sum }_{k=1}^n{D}_k)-(M_0+{\sum }_{k=1}^{n-1}{D}_k)=D_n$ prove inverse reconstruction. For $n=0$ the sum is empty and equals zero, so the initial value is exactly the prescribed $M_0$. For class representatives these finite equalities hold almost surely.