# A Measure-Theoretic Axiomatisation of Causality

arXiv: https://arxiv.org/abs/2305.17139 · HTML: https://mlsanity.com/paper/2305.17139/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 76 references that are in the map.
- First posted on arXiv 19 May 2023.

## Builds on (most cited first, 28 of 28)

- [Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations](https://mlsanity.com/paper/1811.12359/) (2018)
- [Self-Supervised Learning with Data Augmentations Provably Isolates Content from Style](https://mlsanity.com/paper/2106.04619/) (2021)
- [Causality for Machine Learning](https://mlsanity.com/paper/1911.10500/) (2019)
- [Representation Learning via Invariant Causal Mechanisms](https://mlsanity.com/paper/2010.07922/) (2020)
- [Causal Inference in Natural Language Processing: Estimation, Prediction, Interpretation and Beyond](https://mlsanity.com/paper/2109.00725/) (2021)
- [Weakly supervised causal representation learning](https://mlsanity.com/paper/2203.16437/) (2022)
- [Foundations of Structural Causal Models with Cycles and Latent Variables](https://mlsanity.com/paper/1611.06221/) (2016)
- [Causal Consistency of Structural Equation Models](https://mlsanity.com/paper/1707.00819/) (2017)
- [Abstracting Causal Models](https://mlsanity.com/paper/1812.03789/) (2018)
- [Desiderata for Representation Learning: A Causal Perspective](https://mlsanity.com/paper/2109.03795/) (2021)
- [Approximate Causal Abstraction](https://mlsanity.com/paper/1906.11583/) (2019)
- [A Modification of the Halpern-Pearl Definition of Causality](https://mlsanity.com/paper/1505.00162/) (2015)
- [From Statistical to Causal Learning](https://mlsanity.com/paper/2204.00607/) (2022)
- [Causal models for dynamical systems](https://mlsanity.com/paper/2001.06208/) (2020)
- [From Deterministic ODEs to Dynamic Structural Causal Models](https://mlsanity.com/paper/1608.08028/) (2016)
- [Causal Sufficiency and Actual Causation](https://mlsanity.com/paper/2102.02311/) (2021)
- [Causal Modeling of Dynamical Systems](https://mlsanity.com/paper/1803.08784/) (2018)
- [Causal Inference by String Diagram Surgery](https://mlsanity.com/paper/1811.08338/) (2018)
- [Probabilistic Reasoning across the Causal Hierarchy](https://mlsanity.com/paper/2001.02889/) (2020)
- [Homomorphism Autoencoder -- Learning Group Structured Representations from Observed Transitions](https://mlsanity.com/paper/2207.12067/) (2022)
- [Backtracking Counterfactuals](https://mlsanity.com/paper/2211.00472/) (2022)
- [A Causal Analysis of Harm](https://mlsanity.com/paper/2210.05327/) (2022)
- [Quantifying Harm](https://mlsanity.com/paper/2209.15111/) (2022)
- [Towards a Grounded Theory of Causation for Embodied AI](https://mlsanity.com/paper/2206.13973/) (2022)
- [Causal Modeling With Infinitely Many Variables](https://mlsanity.com/paper/2112.09171/) (2021)
- [Subjectivity, Bayesianism, and Causality](https://mlsanity.com/paper/1407.4139/) (2014)
- [Model Explanations via the Axiomatic Causal Lens](https://mlsanity.com/paper/2109.03890/) (2021)
- [Dilations and information flow axioms in categorical probability](https://mlsanity.com/paper/2211.02507/) (2022)

## Built on by (most cited first, 8 of 8)

- [What is causal about causal models and representations?](https://mlsanity.com/paper/2501.19335/) (2025)
- [Measure-Theoretic Anti-Causal Representation Learning](https://mlsanity.com/paper/2510.18052/) (2025)
- [Algorithmic causal structure emerging through compression](https://mlsanity.com/paper/2502.04210/) (2025)
- [Systems with Switching Causal Relations: A Meta-Causal Perspective](https://mlsanity.com/paper/2410.13054/) (2024)
- [Probabilistic Modelling is Sufficient for Causal Inference](https://mlsanity.com/paper/2512.23408/) (2025)
- [ExDAG: an MIQP Algorithm for Learning DAGs](https://mlsanity.com/paper/2406.15229/) (2024)
- [A fine-grained look at causal effects in causal spaces](https://mlsanity.com/paper/2512.11919/) (2025)
- [Orca: Neural Operators for Causal Reasoning in Continuous Time](https://mlsanity.com/paper/2607.27867/) (2026)

## Abstract (arXiv)

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