Potential outcomes and consistency

Fix a target population with probability law $P$, a real outcome, and a nonempty countable discrete set $\mathcal{A}$ of treatment versions. A treatment version $a\in \mathcal{A}$ is a specified intervention protocol, not just a treatment label. The potential outcome $Y^a$ is the real random variable giving the outcome for a unit in the target population under version $a$. Write $A$ for the observed treatment assignment or receipt, with values in $\mathcal{A}$, and $Y$ for the observed outcome. The observed potential outcome is the measurable random variable

$Y^A={\sum }_{a\in \mathcal{A}}{\mathbf{1}}_{\{A=a\}}{Y}^a,$

where the countable sum has at most one nonzero summand at each sample point. Consistency is the assumption $Y=Y^A$ $P$-almost surely. It links the observed outcome to the potential outcome for the received version; it does not assert that treatment assignment is unconfounded.

The full-data law is the joint law of the potential-outcome family, $A$, any baseline covariate random element $L$, and $Y$. The observed-data law is its marginal law of $(A,L,Y)$. A target population, outcome, intervention protocol, and contrast must be fixed before the associated causal target is specified. If interference between units matters, the version or treatment regime must include the relevant assignment or exposure of other units; the notation $Y^a$ by itself does not assert no interference.