Faithfulness of a law to a DAG

Assume AC, and let a joint law $P$ on the measurable vertex variables of a finite DAG $G$ be given. The law is faithful to $G$ if, for every triple of pairwise disjoint vertex sets $A,B,C$, the conditional independence $\sigma (X_A)\perp \text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\perp \sigma (X_B)\mid \sigma (X_C)$ implies that $C$ d-separates $A$ and $B$ in $G$. This is the converse of the global Markov implication; faithfulness is a separate assumption on the specified law, not a consequence of DAG factorization. No claim about Markov equivalence, skeletons, or unshielded colliders is included. Here $X_A$ is the finite random vector $(X_v{)}_{v\in A}$ with the finite-product sigma-algebra; for $A=\mathrm{\varnothing }$ it is the unique random element of the empty-product singleton.