Inverse probability weighting for a causal mean

Let $Q$ be a probability law for an observed discrete or atomic treatment $A$, a covariate random element $L$, and a real-valued observed outcome $Y$. Fix an atom $a$ with $Q(A=a)>0$, let $\mathcal{G}=\sigma (L)$, and set $I_a={\mathbf{1}}_{\{A=a\}},e_a(L)=E_Q[I_a\mid \mathcal{G}]\mathrm{.}$ Assume $e_a(L)>0$ $Q$-almost surely and $I_{a}Y\mathrm{/}{e}_a(L)\in L^1(Q)$. The inverse-probability-weighted functional at $a$ is ${\mathrm{\Psi }}_{\mathrm{I}\mathrm{P}\mathrm{W}}(Q;a)=E_Q\text{\hspace{0.17em}⁣}\left[\frac{I_{a}Y}{e_a(L)}\right],$ provided the displayed weighted variable is integrable. The outcome weight is $I_a\mathrm{/}{e}_a(L)$, formed from the true population propensity $e_a(L)$, not an estimate.

The instructions are: form the event indicator $I_a$; take its conditional expectation given $\sigma (L)$ to obtain $e_a(L)$; divide $I_{a}Y$ by this positive random variable; and integrate under $Q$. This defines a population functional only. It makes no assertion about finite-sample behavior, estimated propensities, or truncated weights.