  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 \textit{what
happens when one intervenes on a 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 \textit{causal space}, consisting of a probability space along with
a collection of transition probability kernels, called \textit{causal 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.
