Reddit reviews An Introduction to Bayesian Analysis: Theory and Methods (Springer Texts in Statistics)
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Most (all?) rigorous treatments Bayesian methods require a rigorous foundation in probability theory - I think that is self explanatory.
The usual foundation for probability theory is measure theory. So, you can't have rigorous foundation in probability theory without knowing measure theory. There are other foundations, but the vast majority of the time we use measure theory - for example, convergence results like the SLLN use the measure theoretic concept of almost-sure convergence.
So, for instance, I could direct you to be a rigorous Bayesian book - for example, this book - but they will assume you already know things like the martingale convergence theorem, Radon-Nikodym, and Borel-Cantelli which are typically covered in measure-theoretic probability.