  We describe the interface between measure theoretic probability and causal
inference by constructing causal models on probability spaces within the
potential outcomes framework. We find that measure theory provides a precise
and instructive language for causality and that consideration of the
probability spaces underlying causal models offers clarity into central
concepts of causal inference. By closely studying simple, instructive examples,
we demonstrate insights into causal effects, causal interactions, matching
procedures, and randomization. Additionally, we introduce a simple technique
for visualizing causal models on probability spaces that is useful both for
generating examples and developing causal intuition. Finally, we provide an
axiomatic framework for causality and make initial steps towards a formal
theory of general causal models.
