[proofplan] The proof converts the graphical back-door condition into the probabilistic hypotheses of the conditional-exchangeability adjustment formula. The causal semantics give a potential outcome $Y_a$ corresponding to the intervention $do(A=a)$ and give consistency on the event $\{A=a\}$. The [single-world intervention graph adjustment criterion](/theorems/9673) converts the back-door blocking condition into conditional exchangeability of $Y_a$ and $A$ given $Z$. We then apply the point-treatment adjustment formula and identify the interventional law with the law of $Y_a$. [/proofplan] [step:Introduce the intervention outcome and the observational law of the covariates] Let $Y_a:(\Omega,\mathcal F)\to(\mathcal Y,\mathcal E_{\mathcal Y})$ denote the potential outcome generated by the compatible causal model under the intervention $do(A=a)$. Let \begin{align*} \mu_Z:\mathcal E_{\mathcal Z}\to[0,1] \end{align*} be the observational marginal law of $Z$, defined by \begin{align*} \mu_Z(C)=\mathbb P(Z\in C) \end{align*} for each $C\in\mathcal E_{\mathcal Z}$. By the intervention semantics of the compatible causal model, the interventional outcome law is the law of this potential outcome: \begin{align*} \mathbb P(Y\in B\mid do(A=a))=\mathbb P(Y_a\in B) \end{align*} for every $B\in\mathcal E_{\mathcal Y}$. By consistency, the observed outcome agrees with the intervention outcome on the observed treatment stratum: \begin{align*} Y=Y_a\quad\text{a.s. on }\{A=a\}. \end{align*} [/step] [step:Use the back-door criterion to obtain conditional exchangeability] The back-door criterion says that $Z$ blocks every path from $A$ to $Y$ that has an arrow into $A$ and contains no descendant of $A$. These are exactly the graphical hypotheses needed to apply the [citetheorem:9673] to the single-world intervention graph for $do(A=a)$. The theorem applies because the formal statement assumes that this graph represents the joint law of the observed variables and the counterfactual outcome $Y_a$ for the compatible causal model. Therefore, \begin{align*} Y_a\perp\!\!\!\perp A\mid Z. \end{align*} [guided] We need to justify the probabilistic independence assumption that will feed into the adjustment formula. The object whose distribution we want is $Y_a$, the outcome that would be observed if the structural mechanism for $A$ were replaced by the constant value $a$. Since $Y_a$ is a counterfactual outcome, ordinary d-separation in the observational graph $G$ is not by itself enough: $Y_a$ is not an ordinary observed node of $G$. The statement supplies the needed bridge by assuming that the single-world intervention graph construction for $do(A=a)$ represents the joint law of the observed variables and $Y_a$ for the compatible causal model. We now verify the hypotheses of the [citetheorem:9673]. The first hypothesis is that $Z$ contains no descendant of the treatment intervention in the relevant single-world intervention graph. This follows from the back-door criterion assumption that no element of $Z$ is a descendant of $A$ in $G$. The second hypothesis is the graphical blocking condition: $Z$ blocks every path from $A$ to $Y$ that has an arrow into $A$. This is exactly the remaining part of the back-door criterion. Applying the [citetheorem:9673] with treatment node $A$, outcome node $Y$, covariate set $Z$, and intervention value $a$ gives the conditional independence statement \begin{align*} Y_a\perp\!\!\!\perp A\mid Z. \end{align*} This is the conditional exchangeability hypothesis needed for adjustment. [/guided] [/step] [step:Apply conditional exchangeability adjustment at the fixed treatment level] The variables $A$, $Z$, $Y$, and $Y_a$ are measurable random elements on standard Borel spaces, so the regular conditional laws appearing below may be chosen as probability kernels. The conditional treatment kernel satisfies $K_{A\mid Z}(z,\{a\})>0$ for $\mu_Z$-almost every $z$, and the regular conditional outcome kernel gives the observed conditional law \begin{align*} z\mapsto \mathbb P(Y\in B\mid A=a,Z=z) \end{align*} for $\mu_Z$-almost every $z$ and every $B\in\mathcal E_{\mathcal Y}$. The preceding steps give the hypotheses of the [point-treatment adjustment formula under conditional exchangeability](/theorems/9671) [citetheorem:9671]: consistency gives $Y=Y_a$ a.s. on $\{A=a\}$, conditional exchangeability gives $Y_a\perp\!\!\!\perp A\mid Z$, and positivity gives the required conditional law on the support of $Z$. Therefore, for every $B\in\mathcal E_{\mathcal Y}$, \begin{align*} \mathbb P(Y_a\in B)=\int_{\mathcal Z}\mathbb P(Y\in B\mid A=a,Z=z)\,d\mu_Z(z). \end{align*} [/step] [step:Identify the potential-outcome law with the interventional law] Combining the intervention-law identity from the first step with the adjustment identity from the previous step yields \begin{align*} \mathbb P(Y\in B\mid do(A=a))=\int_{\mathcal Z}\mathbb P(Y\in B\mid A=a,Z=z)\,d\mu_Z(z). \end{align*} This holds for every measurable set $B\in\mathcal E_{\mathcal Y}$, because the stated regular conditional outcome kernel defines the observed conditional probabilities at $a$ and the conditional treatment positivity condition holds for $\mu_Z$-almost every $z$. Hence the [back-door adjustment formula](/theorems/9695) follows. [/step]