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.