Rigorous book on causal inference Causal Inference is sort of a branch of statistics that defines a new operator called do. I was wondering if there are suggestions of rigorous mathematical books on the subject. Most of what I’ve seen, for example, does not even mentions measure theory when talking about probability.
UPDATE: I've been searching for math books on this, and haven't found much. So any articles/notes are very welcome also.
 A: I will drift from your question a bit.
Causality is not only about the do-operator. The do-operator is just a tiny interpretation and representation of causality. There are several alternative approaches: the Neyman-Rubin potential outcome interpretation, the Cowles-Haavelmo-Heckman (this is a name I just came up with, they don't have an official name) economics view, Causal decision theory, and many, many more. Some references for just a couple of these are in this link.
I am unsure of whether causality is necessarily a branch of statistics. We could also consider it a branch of philosophy. But as the history of science shows us, some sciences start as a branch of philosophy and then become more independent.
As you pointed out in the comments, and it might be hinted from the paragraphs above, you don't need measure theory to do causality. I don't know measure theory, but I do causality. Depending on the view of causality you are working with, you might need it to a greater or less extent.
Here is a small sample of papers in causality I have read and have a (deeper, if you want) mathematical formalism. These are biased towards the Pearl view of causality, which is precisely the one asked about. At the same time, they reflect distinct areas of research in causality.

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*Bongers, S., Forré, P., Peters, J., & Mooij, J. M. (2021). Foundations of structural causal models with cycles and latent variables. The Annals of Statistics, 49(5), 2885-2915.

*Besserve, M., Shajarisales, N., Schölkopf, B., & Janzing, D. (2018). Group invariance principles for causal generative models. In International Conference on Artificial Intelligence and Statistics (pp. 557-565). PMLR.

*Wang, Y., & Jordan, M. I. (2021). Desiderata for representation learning: A causal perspective. arXiv preprint arXiv:2109.03795.

