SCM, DAGs, adjustment

If $n=0$, use the point mass on the empty-product singleton; if $n=1$, use $K_0(s)$. For $n\ge 2$, fix $s\in S$ and build the product recursively: start with $K_0(s)$, and at stage $j+1$ apply [F1] to the probability measure already constructed on the first $j$ factors and the probability measure $K_j(s)$. This constructs a probability measure on the recursively defined first $j+1$ factors, with the product rectangle formula. Finite induction gives $K(s,\cdot )={⨂}_{i<n}{K}_i(s,\cdot )$ and its formula for every $s$.

For a rectangle $R={\prod }_{i<n}{B}_i$, step 1.1 gives $s\mapsto K(s,R)={\prod }_{i<n}{K}_i(s,B_i)$, a finite product of measurable real functions, hence measurable. Let $\mathcal{D}$ be the product-measurable sets $E$ for which $s\mapsto K(s,E)$ is measurable. The class contains the whole space and all rectangles. If $E\in \mathcal{D}$, then $K(s,E^c)=1-K(s,E)$ is measurable. For pairwise disjoint $E_j\in \mathcal{D}$, countable additivity gives $K(s,{\bigcup }_j{E}_j)={\lim }_{N\to \mathrm{\infty }}{\sum }_{j\le N}K(s,E_j)$, a pointwise limit of measurable functions. Thus $\mathcal{D}$ is a lambda-system. Rectangles form a pi-system and generate the product sigma-algebra by [F2], so [F4] gives $\mathcal{D}={⨂}_{i<n}{\mathrm{T}}_i$. Hence $K$ is a probability kernel.

Any other probability kernel with the rectangle formula agrees with $K(s,\cdot )$ on the generating rectangle pi-system for each fixed $s$. The measures are finite, so uniqueness [F3] gives equality on the full product sigma-algebra at every $s$. This proves uniqueness, including the empty product case.

For fixed parent values $x_{\mathrm{pa}(v)}$, the map $u\mapsto f_v(x_{\mathrm{pa}(v)},u)$ is measurable, so its pushforward of ${\nu }_v$ is a probability measure. For each measurable $B\subseteq S_v$, $K_v(x,B)$ is the integral of the jointly measurable indicator $1_B(f_v(x,u))$ against the constant probability kernel $x\mapsto {\nu }_v$; [F2] makes this evaluation measurable in $x$. Hence $K_v$ is a probability kernel, including when the parent set is empty.

Suppose $n\ge 1$. By [F1], the joint law of the exogenous vector is ${⨂}_{j=1}^n{\nu }_{v_j}$. For $1\le r\le n$, let $X_{<r}=(X_{v_1},\dots ,X_{v_{r-1}})$, with $X_{<1}$ the constant empty tuple. Fix a bounded measurable function $g$ of $X_{<r}$ and a measurable set $B\subseteq S_{v_r}$. The recursive construction makes the earlier coordinates functions only of $U_{v_1},\dots ,U_{v_{r-1}}$. The integrand $g(X_{<r})1_{\{X_{v_r}\in B\}}$ does not depend on later exogenous coordinates, whose probability factors integrate to one. The finite product law of the first $r$ exogenous coordinates and Fubini on $({\prod }_{j<r}{E}_{v_j})\times E_{v_r}$ therefore permit integration first in $u_{v_r}$ while fixing the earlier coordinates, giving $E[g(X_{<r})1_{\{X_{v_r}\in B\}}]=E[g(X_{<r})K_{v_r}(X_{\mathrm{pa}(v_r)},B)]\mathrm{.}$ For $r=1$, the earlier product is the singleton and $g$ is constant; the parent tuple is empty. For every $r$, all parent coordinates occur in $X_{<r}$ by the topological order. Thus the displayed identity gives the conditional-law testing identity needed below.

If $n=0$, the endogenous vector is the empty tuple and its law is the point mass on the empty-product singleton, as stated. Suppose $n\ge 1$. The law ${\mu }_1$ of $X_{v_1}$ is $K_{v_1}$, since its parent tuple is empty and step 1.2 applies. Inductively, suppose ${\mu }_{r-1}$ is the law of $X_{<r}$, for some $2\le r\le n$. Let ${\pi }_r$ be the coordinate projection from the preceding product to the parent-coordinate product; it is measurable because preimages of measurable rectangles are coordinate cylinders in the recursive finite product sigma-algebra [F7]. Pull back $K_{v_r}$ to the probability kernel ${\bar{K}}_r(x,B)=K_{v_r}({\pi }_r(x),B)$ on that same source. The identity Dirac kernel $D_r(x,C)=1_C(x)$ and ${\bar{K}}_r$ have common source. By [F5] their product is a probability kernel ${\tilde{K}}_r$ into the preceding product times $S_{v_r}$, with rectangle values ${\tilde{K}}_r(x,C\times B)=1_C(x)K_{v_r}({\pi }_r(x),B)$. Compose it with ${\mu }_{r-1}$, regarded as a probability kernel from a singleton, using [F3]; this gives a probability measure ${\mu }_r$ satisfying ${\mu }_r(C\times B)={\int }_C{K}_{v_r}({\pi }_r(x),B){\mu }_{r-1}(dx)\mathrm{.}$ By step 1.2, the actual law of $(X_{<r},X_{v_r})$ has these same values on every measurable rectangle. Both are probability measures, and the rectangles form a generating pi-system whose whole-space member gives a finite common exhaustion, so [F6] makes the measures equal. Induction gives the displayed iterated topological composition as the full-joint law.

Under $do(X_I=a)$ with $n\ge 1$, the coordinates in $I$ are constants and their conditional kernels are ${\delta }_{a_v}$; every other equation still uses the same independent exogenous coordinate and the same parent values, with an intervened parent replaced by its fixed constant. At an intervened vertex the conditional-law identity is immediate because the coordinate is the fixed value $a_v$; at every other vertex the Fubini calculation of step 1.2 applies with the intervened parent coordinates fixed at their assigned values. Repeating the extension construction of step 2.1 with these kernels proves the interventional product. For $n=0$ the only intervention is empty and the law remains the point mass already described. The observed law is the pushforward under coordinate projection; integrating out the latent coordinates is marginalization and creates no asserted observed-only DAG factorization.

Among active routes with these endpoints choose one of minimum length. If a repeated vertex occurs at an endpoint, discard the initial segment through the later occurrence of $x$, or the terminal segment from the earlier occurrence of $y$. Endpoints are outside $C$, and all retained internal occurrences have their old incidences, so the shortened route is active. Consequently any repetition in a minimum route consists of two internal occurrences of a vertex $v$. Delete the segment between two such occurrences and identify them. Only the two incidences at the joined occurrence of $v$ change. If $v\in C$, its old occurrences were both colliders, so the joined occurrence is still a collider. If $v\mathrm{\notin }C$ and the joined occurrence is a noncollider, it is allowed; if it is a collider and $v$ is an ancestor of $C$, it is also allowed.

The sole remaining possibility is that the join makes $v$ a collider although $v$ is not an ancestor of $C$. Both original occurrences were then noncolliders. The retained edge before the first occurrence points into $v$, so its deleted departure edge must point out of $v$; likewise the deleted arrival edge at the second occurrence points out of $v$, since its retained departure edge points into $v$. Traverse the deleted segment starting with its arrow out of $v$. There must be a first place where its arrows cease to point forward: the final arrow points backward, and an entirely forward return to $v$ would also contradict acyclicity. At that first change is a collider $w$, reached from $v$ by a directed segment. A directed segment in a DAG has no repetitions. Activity of the original route makes $w$ an ancestor of $C$, hence $v$ an ancestor of $C$, a contradiction. Thus every loop deletion above is active and strictly shorter, contradicting minimality. The minimum route is therefore a path.

Suppose an active path joins $X$ to $Y$. From any noncollider follow an outgoing path edge forward until reaching an endpoint or a collider. The reached endpoint is in $X\cup Y$, while a reached collider is an ancestor of $C$ by activity. Thus every path vertex is in $H={\mathrm{An}}_G(X\cup Y\cup C)$. Replace each collider segment $u\to w\leftarrow v$ by its moral edge $u-v$; colliders cannot be consecutive along a path. The resulting walk avoids $C$, since all retained internal vertices were noncolliders and the endpoints are outside $C$. Loop deletion in this undirected graph yields a moral path avoiding $C$.

Conversely suppose a moral path avoiding $C$ joins $X$ to $Y$. Keep its skeleton edges, and expand each remaining moral edge $u-v$ using a common child $w\in H$ as $u\to w\leftarrow v$. This is a finite route; its inserted occurrences are colliders and all its other occurrences are outside $C$. Every collider is in $H$. Call a collider occurrence bad if its vertex is not an ancestor of $C$. Among routes joining $X$ to $Y$ with every vertex in $H$ and every noncollider outside $C$, minimize the number of bad collider occurrences; the expanded route makes the class nonempty. If $w$ is a bad occurrence, it is an ancestor of some $s\in X\cup Y$. A directed path from $w$ to $s$ avoids $C$, since otherwise $w$ would be an ancestor of $C$. If $s\in X$, replace the whole prefix up to this occurrence of $w$ by that directed path traversed backward; if $s\in Y$, replace the suffix by the path traversed forward. If $w=s$, simply terminate at that occurrence. Otherwise the retained incident arrow points into $w$ and the new incident arrow points out of $w$, making it an allowed noncollider. All new internal chain occurrences are noncolliders outside $C$, and other retained occurrences have their original incidences. Intersections of the chain with the retained segment cause repetitions only; no loops are deleted in this step, and activity is defined occurrence by occurrence. At least the selected bad occurrence disappears and no new bad occurrence is introduced, contradicting minimality. Thus the minimum route is active. Its endpoints are distinct because $X,Y$ are disjoint, and [F3] gives an active path with those endpoints.

An active path therefore exists if and only if there is a moral path avoiding $C$. Negating this equivalence proves the claim. If either endpoint set is empty, neither type of path exists and both separation statements hold.

Fix bounded measurable $g(R)$ and set $r=E[g(R)\mid \sigma (C)]$. Conditional independence implies $E[h(L)b(C)g(R)]=E[h(L)b(C)r]$ for every bounded measurable $h,b$: test its conditional-product identity against $b(C)$ and then the defining conditional expectation of $h(L)$. Rectangle indicators generate $\sigma (L,C)$, so [F3] gives $E[Hg(R)]=E[Hr]$ for every bounded $\sigma (L,C)$-measurable $H$. Equivalently $E[g(R)\mid \sigma (L,C)]=r$. All extensions can be made for each fixed $g$: the class of testing events is a lambda-system by complements and disjoint countable unions; bounded versions and monotone convergence justify the sums. No common null set for all functions is needed.

For bounded measurable $h(l,v)$ define $H(l,c)=\int h(l,v)K(l,c,dv)$, bounded and measurable by [F2]. Kernel integration gives $E[b(C)h(L,V)g(R)]=E[b(C)H(L,C)g(R)]=E[b(C)H(L,C)r]=E[b(C)h(L,V)r]$, where the middle identity is step 1.1. Thus $E[h(L,V)g(R)\mid \sigma (C)]=E[h(L,V)\mid \sigma (C)]r$. In taking out bounded measurable conditioning factors here, the event-integral definition first proves the rule for simple factors and [F3] proves it for bounded factors. This proves the first claimed independence. Testing the same calculation with rectangle factors in $(L,V,C)$ and using [F3] also gives $E[g(R)\mid \sigma (L,V,C)]=r$.

Since $r$ is $\sigma (C)$-measurable, [F4] yields $E[g(R)\mid \sigma (C,V)]=r$. For bounded $f(L)$, conditioning first on $\sigma (L,V,C)$ in step 2.1 and then on $\sigma (C,V)$ gives $E[f(L)g(R)\mid \sigma (C,V)]=E[f(L)\mid \sigma (C,V)]r$. The bounded-factor rule justified in step 2.1 applies at both stages. This is the second claimed independence by [F1]. Singleton state spaces and constant kernels obey the same identities.

Put $H={\mathrm{An}}_G(A\cup B\cup C)$. Marginalize vertices outside $H$ in reverse topological order: at removal each has no surviving child, so integration of its kernel gives one. The marginal on $H$ is consequently generated by the original kernels at vertices of $H$, with their original parent sets. Reverse-order integration is elimination of final conditional kernels using [F4], not an interchange of arbitrary dependent integrations.

We prove a stronger finite-graph assertion: if a DAG's full moral graph has a partition of all vertices into $L,R,D$ with no edge from $L$ to $R$, its kernel law satisfies $X_L\perp \text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\perp X_R\mid \sigma (X_D)$. Induct on the number of vertices. For no vertices the variables are constants and the conditional-product identity is immediate. Choose a sink $v$ (the last vertex in a topological order). Marginalizing $v$ retains all other kernels and leaves the DAG with $v$ deleted; its moral graph is a subgraph of the former moral graph, so the remaining partition obeys the induction hypothesis. The original joint law is this marginal extended by $K_v(X_{\mathrm{pa}(v)},d{x}_v)$: no other kernel depends on $x_v$, as $v$ is a sink. This identity follows first on rectangles from the iterated composition, and then on their generated sigma-algebra by finite-measure uniqueness.

If $v\in L$, all its parents are in $(L\setminus \{v\})\cup D$, since a parent in $R$ would give a forbidden skeleton edge. Apply the first conclusion of [F3] to the induction hypothesis with left vector $X_{L\setminus \{v\}}$, right vector $X_R$ and separator $X_D$, extending the left vector by $X_v$. This proves the assertion. If $v\in R$, symmetry gives the same result. If $v\in D$, its parents outside $D\setminus \{v\}$ lie wholly in $L$ or wholly in $R$: two parents on opposite sides would be married by a forbidden moral edge. Choose that side (either side when there are no such parents). The induction hypothesis gives independence of $X_L,X_R$ given $X_{D\setminus \{v\}}$, and the second conclusion of [F3], applied to $K_v$, gives independence given $(X_{D\setminus \{v\}},X_v)=X_D$. Thus even a conditioned vertex's kernel is retained as a kernel extension before conditioning. This completes all cases of the induction.

By [F1], the moral graph on $H$ with $C$ deleted has no component meeting both $A$ and $B$. Let $L$ be the union of components meeting $A$, and $R=H\setminus (L\cup C)$. Then $B\subseteq R$, $A\subseteq L$, and no moral edge joins $L$ to $R$. Steps 1.2 and 2.1 applied to the marginal law of step 1.1 give independence of $X_L,X_R$ given $X_C$. Restrict its bounded-function identity to functions of $X_A,X_B$ to obtain the claim. Empty endpoint sets yield constant functions and satisfy the same identity. No density, positivity, faithfulness, or regular conditional distribution on an arbitrary measurable space is used.

Fix finitely many intervention values $F$. Build one finite graph $J$ with the factual ancestors of $T\cup C$ and, for each $t\in F$, the equation ancestors needed for $X_W^t$ after replacing every $T$-input by its constant $t$. Omit those constant treatment nodes from the response graph. Share every nondescendant of $T$ across the factual and response systems; copy every remaining nonintervened response vertex separately for each $t$. Sharing is legitimate because nondescendants have no treatment ancestor and hence their equations and values are unchanged by intervention, by topological induction. Add one root $U_v$ per used original equation, shared by all copies of that equation. Every endogenous graph vertex uses its original structural function, with constants substituted for treatment parents in the response system, so deterministic point-mass kernels together with the independent root laws give a topological kernel factorization of the joint factual/response law by [F1]. The graph is acyclic: roots come first and each endogenous copy follows its original topological rank. Restrict to ancestors in $J$ of factual $T,C$ and response $W$; this retains the same endpoint law.

Write $G^{\mathrm{\prime }}=G_{\underset{‾}{T}}$ and $H={\mathrm{An}}_{G^{\mathrm{\prime }}}(T\cup W\cup C)$. Every nonroot vertex retained in $J$ projects to an original vertex in $H$. For a response ancestor, its directed route to $W$ has no treatment vertex, because treatment inputs were constants. For a factual ancestor of $C$, its route has no treatment vertex by the nondescendant hypothesis. For a factual ancestor of $T$, truncate its directed route at the first treatment vertex: this has no outgoing treatment arrow. Each of these gives ancestry in $G^{\mathrm{\prime }}$. Moreover, if a factual retained vertex $v\mathrm{\notin }T$ is a descendant of $T$, it cannot be an ancestor of $C$, so there is a directed route from $v$ to a treatment vertex, truncated at its first treatment occurrence. This route lies in $G^{\mathrm{\prime }}$ and avoids $C$, since every vertex on it is a descendant of $T$. Hence $v$ is in a connected component meeting $T$ in the moral graph on $H$ with $C$ deleted.

By the backdoor hypothesis and [F5], no component of that original moral graph meets both $T$ and $W$. Assign each nonroot vertex of $J$ outside $C$ to the component of its projected original vertex. All parent-child and parent-parent moral edges of $J$ outside $C$ preserve the two sides, as follows. A response family has no treatment parent, so its projected nonroot family is a clique already present in the moral graph of $G^{\mathrm{\prime }}[H]$. A factual family without a treatment parent is likewise such a clique. If a factual family has a treatment parent, its child is a factual descendant of $T$ retained as an ancestor of $T$, hence is in a component meeting $T$ by step 2.1. Every nonconditioned nontreatment parent is joined to that child by an arrow retained in $G^{\mathrm{\prime }}$; every treatment parent is itself in a component meeting $T$. These family vertices therefore all lie on the treatment side (although the treatment components need not be the same). This exhausts skeleton edges and moral edges between nonroot vertices.

Assign a root $U_v$ with $v\mathrm{\notin }C$ to the side of its projected child $v$: all copies of $v$ have that same projection. Its moral neighbors are its equation children and their other parents, and the family analysis in step 3.1 puts every neighbor outside $C$ on the same side. If $v\in C$, the child is a single shared nondescendant node, not copied, and has no treatment parent. Its nonconditioned original parents form a clique in $G^{\mathrm{\prime }}[H]$. Assign $U_v$ to their common component side when there are any such parents, and arbitrarily otherwise; after $C$ is deleted it has no neighbor on the opposite side. There are no root-root moral edges, since each equation uses one exogenous root. Let the left side consist of the components meeting $T$ and roots assigned to that side, and let the right side be everything else outside $C$. Thus deleting $C$ from the full moral graph of the retained graph $J$ leaves no edge between the sides. Factual $T$ is on the left and every response $W$ is on the right. In particular no ancestral moral path connects these endpoint sets, and [F5] gives their d-separation in $J$. By its kernel factorization and [F2], $X_T$ is conditionally independent of the finite vector $(X_w^t{)}_{w\in W,t\in F}$ given $X_C$. This argument handles shared roots and shared nondescendants through moral families and does not assume that a projected copied path preserves collider status.

Fix a bounded nonnegative measurable $f(X_T)$. Response events depending on finitely many coordinates form a pi-system generating ${\mathcal{R}}_W^T$. Step 4.1 gives the conditional-product identity for their indicators. The response events for which it holds form a lambda-system: complements subtract the identity for the whole space, and disjoint countable unions follow by summing conditional-expectation versions for the pieces and testing the increasing partial sums over events in $\sigma (X_C)$; ordinary monotone convergence gives a version for the union. Dynkin's theorem [F4] gives the identity for every response event. Simple approximation and monotone convergence extend it to all bounded nonnegative response-measurable functions. Decompose bounded real functions into positive and negative parts to obtain all bounded functions on both sides, hence the claimed independence by [F3]. This cylinder argument applies to any cardinality of the intervention-value set, with no common null set chosen across that set. The fixed-value conclusion is a restriction to one response coordinate vector.

By [F1], $Y^a$ and $A$ are conditionally independent given $L$. By [F2] take regular conditional laws of $Y^a$ given $L$ and given $(A,L)$. The conditional-independence identity implies that, for $P_{A,L}$-almost every $(a^{\mathrm{\prime }},l)$, the latter law equals the former law at $l$. Since $a$ is an atom with $e_a(l)>0$ for $P_L$-almost every $l$, the measure $P_{L\mid A=a}$ is equivalent to $P_L$: its density with respect to $P_L$ is $e_a(l)\mathrm{/}P(A=a)$, which is positive and integrates to one. Hence the conditional-law equality at $A=a$ holds for $P_L$-almost every $l$.

On the event $A=a$, the factual structural recursion and the recursion with the $A$ equation replaced by $a$ agree at $A$. Induction down the fixed topological order shows that they agree at every descendant, in particular $Y=Y^a$. Therefore the conditional law of $Y^a$ given $A=a,L=l$ equals that of $Y$ given $A=a,L=l$ for $P_L$-almost every $l$. Combining with step 1.1 identifies the conditional law of $Y^a$ given $L=l$ with the observed conditional law in the formula.

Disintegrating the joint law of $(Y^a,L)$ by [F2] gives $P(Y^a\in B)=\int P(Y^a\in B\mid L=l)P_L(dl)$. Substitute the equality from step 2.1 to obtain the displayed formula. If two SCM parameter values induce the same observed law, the integrand kernel and $P_L$ agree outside their common null sets; positivity makes those sets null for the target integral. Hence the right side, and so the estimand, is constant on each observational-equivalence class by [F4]. The real-valued expectation formula follows by applying the distribution identity to bounded truncations of $Y$ and using integrability to pass to the limit.

Let $I_A={\mathrm{An}}_G(A)$, including $A$. By recursive substitution, $A$ is measurable with respect to $U_{I_A}$; under $do(A=a)$, the equation at $A$ is replaced, so $M^a$ is measurable with respect to $U_{I_M}$. Under $do(M=m)$, the equation at $M$ is replaced, so $Y^m$ is measurable with respect to $U_{I_Y}$. These claims follow by induction in the finite topological order: an endogenous value depends only on the exogenous coordinates at vertices with directed paths to it in the relevant mutilated graph.

Suppose $v\in I_M\cap I_A$. Since $v\mathrm{\ne }A$, there are directed paths $v\to \cdots \to A$ in $G$ and $v\to \cdots \to M$ in $G_{\bar{A}}$; the latter avoids $A$. Take the last common vertex $w$ of these paths in a topological ordering. Their suffixes are disjoint, so they form a simple fork path $A\leftarrow \cdots \leftarrow w\to \cdots \to M$ with no internal collider. It is retained in $G_{\underset{‾}{A}}$, because the path uses no arrow out of its endpoint $A$, and is active given the empty set. This contradicts (ii), so $I_M\cap I_A=\mathrm{\varnothing }$.

Suppose $v\in I_M\cap I_Y$. There are directed paths $v\to \cdots \to M$ in $G_{\bar{A}}$ and $v\to \cdots \to Y$ in $G_{\bar{M}}$. The first path avoids $A$. The second avoids $M$; it also avoids $A$, since otherwise its segment $A\to \cdots \to Y$ would be a directed $A$-to-$Y$ path avoiding $M$, contrary to (i). Take the last common vertex $w$ of the paths. Their suffixes form a simple fork path $M\leftarrow \cdots \leftarrow w\to \cdots \to Y$ with no internal collider and no internal $A$. It lies in $G_{\underset{‾}{M}}$ and is active given $\{A\}$, contradicting (iii). Thus $I_M\cap I_Y=\mathrm{\varnothing }$.

Fix $v\in I_Y$. There is a directed path $q$ from $v$ to $Y$ in $G_{\bar{M}}$. It avoids $M$ by definition and avoids $A$, because an occurrence of $A$ would leave a directed $A$-to-$Y$ suffix avoiding $M$, contrary to (i). Thus $q$ is retained in $G_{\underset{‾}{M}}$. If an active path $p$ from $M$ to $v$ given $\{A\}$ existed there, choose the last vertex $w$ of $p$ that also lies on $q$ when $q$ is traversed from $v$ to $Y$. Keep the prefix of $p$ from $M$ to $w$ and append the directed suffix of $q$ from $w$ to $Y$. The suffix after $w$ is disjoint from the prefix by the choice of $w$, so this is a path. Its old internal vertices remain active; $w\mathrm{\notin }A$ and the outgoing directed edge makes it a noncollider; all new internal vertices are noncolliders outside $A$. This would be an active path from $M$ to $Y$ given $A$, contradicting (iii). Hence $A$ d-separates $M$ and $v$ in $G_{\underset{‾}{M}}$. In the expanded graph each added root has only one skeleton neighbor, its own equation vertex, so a simple path cannot contain a root internally. An active path ending at $U_v$ would therefore have penultimate vertex $v$; deleting its last edge leaves an active path from $M$ to $v$ (or $v=M$). The latter case is excluded by $M\mathrm{\notin }{I}_Y$, and the former was just excluded. Adding roots does not create new directed descendants of endogenous vertices, so the remaining collider-activity tests are unchanged. Thus $A$ also d-separates $M$ and the whole root set $U_{I_Y}$ in the expanded graph with arrows out of $M$ deleted.

By [F3], the disjoint exogenous vectors $U_{I_M}$ and $U_{I_A}$ are independent. Step 1.1 makes $M^a$ and factual $A$ measurable functions of these respective vectors, so $M^a\perp A$.

The finite vectors $U_{I_M}$ and $U_{I_Y}$ are independent by [F3]. By step 1.1, $M^a$ and the response family $(Y^m{)}_{m\in S_M}$ are measurable functions of these disjoint vectors; in particular $M^a\perp U_{I_Y}$.

Adjoin all exogenous coordinates as independent root vertices. In the factual law, no strict descendant of $M$ is an ancestor of $(M,A,U_{I_Y})$: it cannot be an ancestor of an exogenous root, and if it were an ancestor of $A$, the directed path from $M$ to $A$ would remain in $G_{\underset{‾}{A}}$ and contradict (ii). Reverse-topological integration therefore marginalizes all other vertices and leaves the joint law of $(M,A,U_{I_Y})$ factored on its ancestral subgraph, which has no outgoing edge from $M$. This is a subgraph of the expanded mutilated graph in step 1.4, so the separation there still holds. The global Markov implication [F2], interpreted by the conditional-independence definition [F4], gives $M\perp U_{I_Y}\mid A$.

In $G_{\bar{M}}$ there is no directed path from $A$ to $Y$, since every directed $A$-to-$Y$ path in $G$ contains $M$ and arrows into $M$ were deleted. With $M$ fixed, changing the equation at $A$ therefore cannot change $Y$ by recursive substitution: any directed route by which $A$ could affect $Y$ would have to reach the fixed node $M$. Under $do(A=a)$, the mediator equals $M^a$. Replacing the mediator equation by that same realized value leaves every other equation unchanged; the no-directed-path argument shows that leaving $A$ factual has no further effect on $Y$. Thus $Y^a=Y^{M^a}$.