Conditional-independence splice lemma

Apply [F1] to the coordinate pair on [F1, F2] $(S_2\times S_3,{\mu }_{23})$ and then [F2]. This gives a probability kernel $Q:S_2\to S_3$ such that, for $A_2,A_3$ measurable, ${\mu }_{23}(A_2\times A_3)={\int }_{A_2}Q(s_2,A_3){\mu }_2(d{s}_2)\mathrm{.}$ Choice is used precisely by [F1]--[F2] to select and factor the conditional law.

Lift $Q$ to the kernel [F3, F4, F7, step 1.1] $\tilde{Q}((s_1,s_2),\cdot )=Q(s_2,\cdot )$ and, for a measurable $C\subseteq S_1\times S_2\times S_3$, set $\mu (C)={\int }_{S_1\times S_2}Q(s_2,C_{(s_1,s_2)}){\mu }_{12}(d(s_1,s_2))\mathrm{.}$ Each section $C_{(s_1,s_2)}$ is measurable by [F7], and the integrand is measurable by [F3], applied to $1_C$ and $\tilde{Q}$. For disjoint $C_j$, their sections are disjoint and [F4] passes the increasing partial sums through the outer integral. Thus $\mu$ is countably additive; $\mu (\mathrm{\varnothing })=0$ and $\mu (S_1\times S_2\times S_3)=1$. Hence it is a probability measure, including when one of the displayed test sets below is empty.

Taking $C=A_1\times A_2\times S_3$ in step 2.1 gives [F6, step 1.1, step 2.1] $\mu (C)={\mu }_{12}(A_1\times A_2)$. Taking $C=S_1\times A_2\times A_3$ and using step 1.1 gives $\mu (C)={\mu }_{23}(A_2\times A_3)$. The rectangle pi-systems and [F6] therefore identify both required marginals on their full product sigma-algebras.

For bounded measurable $g:S_3\to \mathbb{R}$, write [F3, F5, step 1.1, step 2.1, step 3.1] $Qg(s_2)=\int g(s_3)Q(s_2,d{s}_3)$. The construction in step 2.1, first for indicators, then for simple functions, and then for positive and negative parts, shows ${\mathbb{E}}_{\mu }[g(S_3)\mid \sigma (S_1,S_2)]=Qg(S_2)\mathrm{.}$ The $(2,3)$ marginal and step 1.1 likewise show ${\mathbb{E}}_{\mu }[g(S_3)\mid \sigma (S_2)]=Qg(S_2)$. Criterion [F5] now gives $S_1\perp \text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\perp S_3\mid S_2$.

Let $\nu$ be any other law with the two marginals and the stated [F5, F6, step 2.1, step 3.1] conditional independence. Its $(2,3)$ marginal makes $Qg(S_2)$ a conditional expectation of $g(S_3)$ given $S_2$; [F5] then makes it the conditional expectation given $(S_1,S_2)$. Consequently, for every measurable rectangle, $\nu (A_1\times A_2\times A_3)=\int 1_{A_1}(s_1)1_{A_2}(s_2)Q(s_2,A_3)d{\mu }_{12},$ which equals the value of $\mu$ from step 2.1. Rectangles form a pi-system containing the whole space, so [F6] gives $\nu =\mu$.