Positivity for discrete treatment

Let $A$ take values in a countable discrete set on a target probability space, and let $L$ be the baseline covariate random element. For a treatment value $a$ of interest, the conditional treatment probability is the almost-sure class

$e_a=E[{\mathbf{1}}_{\{A=a\}}\mid \sigma (L)],$

using the conditional-expectation convention of Conditional expectation as an ae class under the The Axiom of Choice. Positivity for $a$ on the target population means both $P(A=a)>0$ and $e_a>0$ almost surely under the target law of $L$. Positivity for a contrast is required at every treatment value in that contrast.

The almost-sure condition is relative to the target population: values of $L$ with zero target probability impose no condition. Strict positivity does not mean that $e_a$ is bounded away from zero. The marginal condition makes the observed treatment-stratum law $P(\cdot \mid A=a)$ available.