An ANM density factorization implies conditional independence under d-separation

For Borel noise densities use the displayed product. Otherwise [F1] supplies positive finite Borel versions under the assumed Countable Choice; use their Borel product throughout the integrations below. It has the same observational law, and its marginal densities agree almost everywhere with the stipulated marginal versions, so the final almost-everywhere identity transfers to those versions. Put $H={\mathrm{An}}_G(A\cup B\cup C)$. No vertex outside $H$ is a parent of a vertex in $H$, since that would make it an ancestor of $A\cup B\cup C$. Integrate the product density [F1] over $V\setminus H$ in reverse topological order. At each elimination, all child factors have already been integrated and no child lies in $H$, so the only remaining factor involving the current coordinate $x_v$ is $q_v(x_v-f_v(x_{\mathrm{pa}(v)}))$. Its section integral is one by [F2]. Tonelli [F3] justifies each section integral; consequently the marginal density on $H$ is ${\prod }_{v\in H}{q}_v(x_v-f_v(x_{\mathrm{pa}(v)}))$.

By [F4], deleting $C$ from the moral graph on $H$ leaves no path from $A$ to $B$. Partition its connected components into $L$, the union of components meeting $A$, and $R$, the union of the remaining components. Then $A\subseteq L$, $B\subseteq R$, and there is no moral edge between $L$ and $R$. For each $v\in H$, the factor indexed by $v$ in step 1.1 depends only on $\{v\}\cup \mathrm{pa}(v)\subseteq H$; this set is a clique in that moral graph by [F5]. Hence no such factor involves both an $L$-coordinate and an $R$-coordinate. Assign factors involving an $L$-coordinate to $U$, and assign all other factors to $W$. Their finite products give a factorization of the density on $H$ as $U(x_L,x_C)W(x_R,x_C)$.

Integrate the factorization in step 2.1 over the hidden coordinates $L\setminus A$ and $R\setminus B$. Since the factors separate and are nonnegative, Tonelli [F3] yields measurable functions $u(x_A,x_C)$ and $w(x_B,x_C)$ such that $p_{A\cup B\cup C}(x_A,x_B,x_C)=u(x_A,x_C)w(x_B,x_C)$ almost everywhere. Each function is strictly positive because it integrates a strictly positive factor over its hidden coordinates; if that coordinate set is empty, it is the original strictly positive factor.

Define $\alpha (x_C)=\int u(x_A,x_C)d{x}_A$ and $\beta (x_C)=\int w(x_B,x_C)d{x}_B$. Tonelli makes these measurable and gives $p_{A\cup C}=u\beta$ and $p_{B\cup C}=w\alpha$ almost everywhere. Also $p_C(x_C)=\int u(x_A,x_C)w(x_B,x_C)d{x}_{A}d{x}_B$ for almost every $x_C$. The marginal $p_C$ is positive, since it integrates the everywhere-positive density over any unit box in the remaining coordinates, and it is finite almost everywhere since $\int p_C=1$. For almost every such $x_C$, strict positivity of $u,w$ gives $\alpha ,\beta >0$. If, for example, $\alpha =\mathrm{\infty }$, then $\beta >0$ supplies a set of $b$-values of positive measure where $w(b,x_C)>1\mathrm{/}n$ for some $n$, so Tonelli makes $\int u(a,x_C)w(b,x_C)dadb=\mathrm{\infty }$; the same argument applies if $\beta =\mathrm{\infty }$. Hence both integrals are finite almost everywhere, Tonelli factors the last integral as $p_C=\alpha \beta$, and multiplying the finite marginal identities gives the claimed conditional-density factorization.