A Measure-Theoretic Axiomatisation of Causality

Causality is a central concept in a wide range of research areas, yet there is still no universally agreed axiomatisation of causality. We view causality both as an extension of probability theory and as a study of $\mathit{\text{what}}\mathit{\text{happens}}\mathit{\text{when}}\mathit{\text{one}}\mathit{\text{intervenes}}\mathit{\text{on}}\mathit{\text{a}}\mathit{\text{system}}$, and argue in favour of taking Kolmogorov’s measure-theoretic axiomatisation of probability as the starting point towards an axiomatisation of causality. To that end, we propose the notion of a $\mathit{\text{causal}}\mathit{\text{space}}$, consisting of a probability space along with a collection of transition probability kernels, called $\mathit{\text{causal}}\mathit{\text{kernels}}$, that encode the causal information of the space. Our proposed framework is not only rigorously grounded in measure theory, but it also sheds light on long-standing limitations of existing frameworks including, for example, cycles, latent variables and stochastic processes.