Doob decomposition of an integrable adapted process

First repair the integral foundation inherited by RN and conditional expectation. Augment any finite disjoint display of a nonnegative simple function by the complement with coefficient $0$. Intersections of two augmented displays partition the whole space and have equal coefficients wherever nonempty, so finite additivity and $0\cdot (+\mathrm{\infty })=0$ prove representation independence. Common refinements give simple monotonicity and additivity; handle scalar $0$ directly and positive scalars termwise. Supremum over simple minorants and the sets $\{f_j\ge cs\}$, $0<c<1$, give monotone convergence; increasing simple approximations then give nonnegative additivity. Positive/negative and real/imaginary decompositions give finite $L^1$ linearity. With these facts substituted at the affected foundation, the cited RN proof gives its density, and its event-integral existence and uniqueness argument gives the conditional-expectation class and algebra in [F1], [F4] and [F5]. Each $V_k=X_k-X_{k-1}$ is therefore real measurable and integrable, since $E\mathrm{\mid }{V}_k\mathrm{\mid }\le E\mathrm{\mid }{X}_k\mathrm{\mid }+E\mathrm{\mid }{X}_{k-1}\mathrm{\mid }$. AC permits choosing a finite real integrable ${\mathcal{F}}_{k-1}$-measurable version $a_k$ of its conditional expectation for every $k\ge 1$. Set $A_0=0$, $A_n={\sum }_{k=1}^n{a}_k$, and $M_n=X_n-A_n$. For $k\le n$ one has ${\mathcal{F}}_{k-1}\subseteq {\mathcal{F}}_{n-1}$; therefore $A_n$ is predictable for $n\ge 1$. Finite sums and differences show that $A$ is integrable and $M$ is adapted and integrable.

The increment $M_n-M_{n-1}=V_n-a_n$ has conditional expectation $a_n-a_n=0$ given ${\mathcal{F}}_{n-1}$, by linearity and known-variable conditioning. Since $M_0=X_0$ is integrable and ${\mathcal{F}}_0$-measurable, [F6] makes $M$ a martingale. The displayed decomposition holds pointwise for the chosen representatives.

If $X=\tilde{M}+\tilde{A}$ is another normalized decomposition, then ${\tilde{A}}_n-{\tilde{A}}_{n-1}$ is integrable and ${\mathcal{F}}_{n-1}$-measurable: for $n=1$ use ${\tilde{A}}_0=0$, and for $n>1$ use predictability and nesting. Conditioning the decomposition increment and using the zero martingale drift gives $E[V_n\mid {\mathcal{F}}_{n-1}]={\tilde{A}}_n-{\tilde{A}}_{n-1}$ a.s. Thus these increments equal $a_n$ a.s. Induction from zero gives ${\tilde{A}}_n=A_n$ and then ${\tilde{M}}_n=M_n$ a.s. for each $n$. The sets where either equality fails are ambient measurable null sets; their countable union is null by [F7]. Off that one set both entire sequences agree. No completeness of the filtration is used.