Conditional-class standardization identifies a causal mean

Fix $P$ in the stated model class, let $I_a={\mathbf{1}}_{\{A=a\}}$, $e_a=E_P[I_a\mid \mathcal{G}]$, ${\mu }_a=E_P[Y^a\mid \mathcal{G}]$, and $q_a=E_P[I_{a}Y\mid \mathcal{G}]$. By [F4] and [F10], $P(A=a)=E_P[I_a]=E_P[e_a]$. Global consistency, or the stated atom-consistency directly, gives $I_{a}Y=I_a{Y}^a$ almost surely, so $I_{a}Y$ is integrable. By [F3], full-family exchangeability implies $Y^a\perp \text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\perp A\mid \mathcal{G}$; in the general atom formulation this coordinate independence is assumed directly; [F2] then gives $q_a=E_P[I_a{Y}^a\mid \mathcal{G}]=e_a{\mu }_a$.

Let $\lambda =P_L=Q_L$ be the common law of the covariate coordinate. Since $\mathcal{G}=\sigma (L)=\{L^{-1}(B):B\in \mathcal{S}\}$, each real $\mathcal{G}$-measurable version $V$ factors as $v(L)$: for each rational $r$ choose $B_r\in \mathcal{S}$ with $\{V<r\}=L^{-1}(B_r)$, and put $v(x)=\inf \{r\in \mathbb{Q}:x\in B_r\}$, replacing nonfinite values by $0$. Its rational sublevel sets make $v$ measurable, and $v(L)=V$. Apply this to the five conditional versions, writing ${\tilde{e}}_a,{\tilde{q}}_a$ for the $Q$-conditional classes in [F1], and ${\bar{e}}_a,{\bar{q}}_a,{\bar{\mu }}_a$ for the $P$-conditional classes $e_a,q_a,{\mu }_a$ from step 1.1. By [F12], all five functions are integrable under $\lambda$. For each measurable set $B$ in the covariate space, the defining event integrals for $e_a,q_a$ and the fact that $Q$ is the observed-data law give ${\int }_B{\tilde{e}}_{a}d\lambda =Q(A=a,L\in B)=E_P[I_a{\mathbf{1}}_{\{L\in B\}}]={\int }_B{\bar{e}}_{a}d\lambda ,$ ${\int }_B{\tilde{q}}_{a}d\lambda =E_Q[I_{a}Y{\mathbf{1}}_{\{L\in B\}}]=E_P[I_{a}Y{\mathbf{1}}_{\{L\in B\}}]={\int }_B{\bar{q}}_{a}d\lambda \mathrm{.}$ Each difference is integrable and has zero integral over every $B$, so it and zero are conditional-expectation versions of zero given the full sigma-algebra on $(\mathcal{L},\mathcal{S},\lambda )$. Uniqueness [F5] gives ${\tilde{e}}_a={\bar{e}}_a$ and ${\tilde{q}}_a={\bar{q}}_a$ $\lambda$-almost everywhere. Factoring the a.s. identity in step 1.1 gives ${\bar{q}}_a={\bar{e}}_a{\bar{\mu }}_a$ $\lambda$-almost everywhere; positivity therefore makes the ratio in [F1] equal to ${\bar{\mu }}_a$ $Q_L$-almost everywhere. Thus $m_a$ is integrable and the standardization functional is well-defined.

By [F12] and step 2.1, $E_Q[m_a(L)]=\int m_{a}d\lambda =\int {\bar{\mu }}_{a}d\lambda =E_P[{\mu }_a]$. The tower property [F4] then gives ${\mathrm{\Psi }}_{\mathrm{s}\mathrm{t}\mathrm{d}}(Q;a)=E_P[Y^a]$.

For either potential-outcome formulation in the Statement, let ${\mathrm{\Theta }}_a$ be the class of full-data laws on the fixed measurable spaces for its variables that satisfy the stated integrability, consistency, conditional-exchangeability (or atom-specific independence), and positivity assumptions. Define the target by ${\psi }_a(P)=E_P[Y^a]$ and the observation map by $\mathsf{O}(P)={\mathcal{L}}_P(A,L,Y)=Q$, the pushforward of $P$ under $(A,L,Y)$. Steps 2.1–3.1 show for every $P\in {\mathrm{\Theta }}_a$ that ${\psi }_a(P)={\mathrm{\Psi }}_{\mathrm{s}\mathrm{t}\mathrm{d}}(\mathsf{O}(P);a)$, where the conditional classes in the right-hand side are determined uniquely by that observed law. Hence the target factors through the observation map on this model, and [F7]–[F9] imply identifiability from $Q$.

Under the additional countably discrete treatment and standard-Borel covariate hypotheses of [F11], the earlier theorem's RCD regression evaluated at $L$ equals ${\mu }_a$ almost surely on the target support, as established in its proof steps 2.1–3.1. Step 2.1 gives the same class for this algorithm's $m_a$. Thus the two standardization functionals coincide. The causal identification on this overlapping domain is the earlier theorem applied to the conditional-class algorithm; steps 1.1–4.1 supply the extension to arbitrary covariate random elements and the general target-atom formulation.