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 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 causal space, consisting of a probability space along with a collection of transition probability kernels, called 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.