  We extend Robins' theory of causal inference for complex longitudinal data to
the case of continuously varying as opposed to discrete covariates and
treatments. In particular we establish versions of the key results of the
discrete theory: the g-computation formula and a collection of powerful
characterizations of the g-null hypothesis of no treatment effect. This is
accomplished under natural continuity hypotheses concerning the conditional
distributions of the outcome variable and of the covariates given the past. We
also show that our assumptions concerning counterfactual variables place no
restriction on the joint distribution of the observed variables: thus in a
precise sense, these assumptions are "for free," or if you prefer, harmless.
