Doob differences inherit coordinate oscillation widths

By [F1] and [F2], the joint law of $(Z_1,\dots ,Z_n)$ is $\mu$. For $1\le i<n$, repeated application of [F4] gives a measurable version $H_i$ of the integral over the unrevealed coordinates; clause 3 of [F4] makes each such section-integral function integrable after assigning $0$ on its exceptional set. Set $H_0=\int fd\mu$ and $H_n=f$, which are integrable by the hypothesis. Let $M_i=H_i(Z_1,\dots ,Z_i)$. Then $M_i$ is integrable and ${\mathcal{F}}_i$-measurable by [F3], [F5], and the product-integral construction; [F6] identifies the expectation of each integrable $M_i$ with its Lebesgue integral.

Fix $i$. Fubini applied to $\mathrm{\mid }f\mathrm{\mid }$ gives a measurable full-measure set of prefixes $p=(x_1,\dots ,x_{i-1})$ for which the integral of $\mathrm{\mid }f(p,x_i,y)\mathrm{\mid }$ over ${\mu }_i\otimes {\mu }_{i+1:n}$ is finite. For such a prefix there is a coordinate value $x_i^0$ whose future-section integral is finite. The coordinate oscillation assumption gives $\mathrm{\mid }f(p,x_i,y)\mathrm{\mid }\le \mathrm{\mid }f(p,x_i^0,y)\mathrm{\mid }+c_i$ for every $x_i,y$, so every future section is integrable and $g_p(x_i):=\int f(p,x_i,y)d{\mu }_{i+1:n}(y)$ is finite. Integrating the same pointwise oscillation bound shows $\mathrm{\mid }{g}_p(x_i)-g_p(x_i^{\mathrm{\prime }})\mathrm{\mid }\le c_i$ for all $x_i,x_i^{\mathrm{\prime }}$.

If $A\in {\mathcal{F}}_{i-1}$, the definition of the sigma-algebra generated by the prefix writes $A=\{(Z_1,\dots ,Z_{i-1})\in C\}$ for a product-measurable $C$. By [F2], [F3], and [F4], integrating $H_i$ first over coordinate $i$ gives the same integral over $A$ as integrating $H_{i-1}$. Hence ${\int }_A(M_i-M_{i-1})d\mathbb{P}=0$.

On the full-measure prefix set from step 1.2, define $u_i(p):=\inf \{r\in \mathbb{R}:{\mu }_i(g_p>r)=0\}$ and ${\mathrm{\ell }}_i(p):=\sup \{r\in \mathbb{R}:{\mu }_i(g_p<r)=0\}$. They are finite because $g_p$ has finite range diameter at most $c_i$. The definitions and [F7] imply $g_p\le u_i(p)$ and $g_p\ge {\mathrm{\ell }}_i(p)$ almost everywhere: the set of almost-everywhere upper bounds is upward closed, so $u_i(p)+1\mathrm{/}m$ belongs to it for every integer $m\ge 1$ by the defining infimum; similarly ${\mathrm{\ell }}_i(p)-1\mathrm{/}m$ is an almost-everywhere lower bound. Thus the explicitly defined sets $\{g_p>u_i(p)+1\mathrm{/}m\}$ and $\{g_p<{\mathrm{\ell }}_i(p)-1\mathrm{/}m\}$ are null. Their countable union is null by [F7], and outside it letting $m\to \mathrm{\infty }$ gives both endpoint inequalities. These sequences and exceptional sets are specified by formulas, with no choice axiom. Also $u_i(p)-{\mathrm{\ell }}_i(p)\le c_i$: otherwise choose $\delta >0$ with $u_i(p)-{\mathrm{\ell }}_i(p)>c_i+2\delta$. The sets $\{g_p>u_i(p)-\delta \}$ and $\{g_p<{\mathrm{\ell }}_i(p)+\delta \}$ then both have positive ${\mu }_i$-measure by the definitions, so a point from each would violate the oscillation bound in step 1.2. For each $q\in \mathbb{R}$, $u_i(p)>q$ iff ${\mu }_i(g_p>q)>0$: if the latter holds, $\{g_p>q\}={\bigcup }_{m\ge 1}\{g_p>q+1\mathrm{/}m\}$ and [F7] give a positive-measure higher level, while $q<u_i(p)$ means $q$ is not an almost-everywhere upper bound. Similarly, ${\mathrm{\ell }}_i(p)<q$ iff ${\mu }_i(g_p<q)>0$, using $\{g_p<q\}={\bigcup }_{m\ge 1}\{g_p<q-1\mathrm{/}m\}$ and [F7]. Repeated Fubini in [F4] makes $(p,x_i)\mapsto g_p(x_i)$ measurable on this set; applying [F4] to the indicators of these level sets makes $p\mapsto {\mu }_i(g_p>q)$ and $p\mapsto {\mu }_i(g_p<q)$ measurable. Thus both endpoints are measurable.

Fubini gives $H_{i-1}(p)=\int g_p(x_i)d{\mu }_i(x_i)$ for almost every prefix, so ${\mathrm{\ell }}_i(p)\le H_{i-1}(p)\le u_i(p)$. Set $A_i(p)={\mathrm{\ell }}_i(p)-H_{i-1}(p)$ and $B_i(p)=u_i(p)-H_{i-1}(p)$ there, and set both to $0$ on the exceptional prefix set. These are finite measurable endpoints, contain $0$, and have width at most $c_i$. For ${\mu }_i$-almost every $x_i$, $g_p(x_i)-H_{i-1}(p)$ lies between them; by the product law this is $D_i$ almost surely. Together with step 2.1 this proves the asserted endpoint and event-integral properties for every $i$.