Standardization and IPW identification

Set $I={\mathbf{1}}_{\{A\in C\}}$, $e=E[I\mid \mathcal{G}]$, and $U=E[Z\mid \mathcal{G}]$. These conditional-expectation classes exist under AC, and $0\le e\le 1$ almost surely by positivity and preservation of constants.

For each integer $n\ge 1$, put $Z_n=\max (-n,\min (Z,n))$. The functions $z\mapsto \max (-n,\min (z,n))$ are bounded and measurable, so [F1] applied to $1_C(A)$ and this truncation of $Z$ gives $E[I{Z}_n\mid \mathcal{G}]=eE[Z_n\mid \mathcal{G}]$.

By [F4], $Z_n\to Z$ in $L^1$. Since $\mathrm{\mid }I\mathrm{\mid }\le 1$, $I{Z}_n\to IZ$ in $L^1$; [F3] then gives $E[I{Z}_n\mid \mathcal{G}]\to E[IZ\mid \mathcal{G}]$ in $L^1$ and $eE[Z_n\mid \mathcal{G}]\to eU$ in $L^1$, using $0\le e\le 1$.

The two sides of step 2.1 are equal for every $n$ and converge in $L^1$ to the two sides claimed in the Statement. Uniqueness of limits in $L^1$ gives $E[IZ\mid \mathcal{G}]=eU$ almost surely.

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.

Put $\mathcal{G}=\sigma (L)$, $I_a={\mathbf{1}}_{\{A=a\}}$, $e_a=E_P[I_a\mid \mathcal{G}]$, ${\mu }_a=E_P[Y^a\mid \mathcal{G}]$, and $H_n=\min (n,1\mathrm{/}{e}_a)$ on $\{e_a>0\}$, setting $H_n=0$ on the null complement. Let $\lambda =P_L=Q_L$ be the common covariate law. 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 versions of the $P$- and $Q$-conditional propensities to obtain functions ${\bar{e}}_a$ and ${\tilde{e}}_a$. By [F13] both are integrable under $\lambda$. For every measurable set $B$ in that space, ${\int }_B{\tilde{e}}_{a}d\lambda =Q(A=a,L\in B)=E_P[I_a{\mathbf{1}}_{\{L\in B\}}]={\int }_B{\bar{e}}_{a}d\lambda \mathrm{.}$ The 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 [F14] gives ${\tilde{e}}_a={\bar{e}}_a$ $\lambda$-almost everywhere. Thus the propensity used by [F1] agrees with the full-data propensity. By [F6] and [F11], $P(A=a)=E_P[I_a]=E_P[e_a]$. Then $H_n$ is bounded and $\mathcal{G}$-measurable, $H_n\uparrow 1\mathrm{/}{e}_a$ almost surely, and $I_{a}Y=I_a{Y}^a$ almost surely by global consistency or the stated atom-consistency. The conditional expectations exist by AC.

Full-family exchangeability, or the stated coordinate independence directly, gives $Y^a\perp \text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\perp A\mid \mathcal{G}$. By [F12], this also holds for $\mathrm{\mid }{Y}^a\mathrm{\mid }$, so [F2] gives $E[I_a\mathrm{\mid }{Y}^a\mathrm{\mid }\mid \mathcal{G}]=e_{a}E[\mathrm{\mid }{Y}^a\mathrm{\mid }\mid \mathcal{G}]$. For each $n$, taking out the bounded factor $H_n$ yields $E[I_a\mathrm{\mid }{Y}^a\mathrm{\mid }{H}_n]=E[e_a{H}_{n}E[\mathrm{\mid }{Y}^a\mathrm{\mid }\mid \mathcal{G}]]$. Monotone convergence [F4] on the left and dominated convergence [F5] on the right, since $0\le e_a{H}_n\uparrow 1$ and $E[\mathrm{\mid }{Y}^a\mathrm{\mid }\mid \mathcal{G}]\in L^1$, give $E[I_a\mathrm{\mid }{Y}^a\mathrm{\mid }\mathrm{/}{e}_a]=E\mathrm{\mid }{Y}^a\mathrm{\mid }<\mathrm{\infty }$. Thus the observed weighted outcome is integrable.

By [F3], [F2] also gives $E[I_a{Y}^a\mid \mathcal{G}]=e_a{\mu }_a$. For each $n$, $E[I_a{Y}^a{H}_n]=E[e_a{H}_n{\mu }_a]$. The left sides converge to $E[I_a{Y}^a\mathrm{/}{e}_a]$ by [F5] with integrable majorant $I_a\mathrm{\mid }{Y}^a\mathrm{\mid }\mathrm{/}{e}_a$ from step 2.1; the right sides converge to $E[{\mu }_a]$ by [F5] with majorant $\mathrm{\mid }{\mu }_a\mathrm{\mid }$. Hence $E[I_a{Y}^a\mathrm{/}{e}_a]=E[{\mu }_a]$.

By step 1.1 choose the common covariate-law representative ${\bar{e}}_a$ and set the observed-coordinate ratio to zero on its $\lambda$-null nonpositive set. The change-of-variables identity [F13], first for the absolute value and then for the integrable signed function, shows that the Q-integrand in [F1] and its P observed-coordinate version have the same integral. Their common absolute integral is $E_P[I_a\mathrm{\mid }{Y}^a\mathrm{\mid }\mathrm{/}{e}_a]$, finite by step 2.1 and $I_{a}Y=I_a{Y}^a$ almost surely, so [F1] is well-defined. The same identity, atom-consistency, step 3.1, and the tower property [F6] give $E_Q[I_{a}Y\mathrm{/}{e}_a(L)]=E_P[I_a{Y}^a\mathrm{/}{e}_a]=E_P[Y^a]$.

Let ${\mathrm{\Theta }}_a$ be the class of full-data laws on the fixed measurable spaces for the variables in the applicable formulation of the Statement that satisfy its integrability, consistency, conditional-exchangeability (or atom-specific independence), and positivity assumptions. Define ${\psi }_a(P)=E_P[Y^a]$ and the observation map $\mathsf{O}(P)={\mathcal{L}}_P(A,L,Y)=Q$, the pushforward of $P$ under $(A,L,Y)$. Step 4.1 proves for every $P\in {\mathrm{\Theta }}_a$ that ${\psi }_a(P)={\mathrm{\Psi }}_{\mathrm{I}\mathrm{P}\mathrm{W}}(\mathsf{O}(P);a)$. This factor is a function of the observed law alone, with its conditional propensity class determined by that law as in step 1.1; [F7]–[F9] therefore give identifiability on this model.