# Bayesian Inference in Measure Theory

What's the deal. How does this work, or can you point me to some references? I tried $\mu(A|B) = \mu(A \cap B) / \mu(B)$ and got stuck on $\mu(B) = 0$.

Edit: Sorry for being lazy. My background is the basics of measure theory (working on it): measurable spaces, measurable functions, Lebesgue integral, that's about it so far. I haven't yet learned much about measure theory and probability. I am mainly just curious if there is a "formula" for Bayes' rule in measure theory? And interested in anything relevant.

One motivation is we often model a game in economics by have a finite set of states of the world with a prior distribution, then we learn that the true state is in some subset and update based on Bayes rule. I haven't seen how to model this with an infinite state space (I can only think of special cases where it would work).

Thanks!

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You can't condition on measure zero sets, it just doesn't make sense. Related question –  icurays1 Jan 25 at 21:38
I don't get it? You want a measure theoretc treatment of Bayesian statistics? –  Michael Greinecker Jan 25 at 21:44
Also, did you mean to say $\mu(A\vert B)=\mu(A\cap B)/\mu(B)$? The equation you have isn't correct. –  icurays1 Jan 25 at 21:44
@usul I fyour interest is probabilitisc conditioning, the most comprehensive book is probably Conditional Measures and Applications by M.M. Rao. A treatment of Bayesian statistics can be found here. None of these is for the beginner though. –  Michael Greinecker Jan 25 at 22:40
@usul Can you specfy what your background is and what you want to learn this for? Then I might give you a more tailormade reference. –  Michael Greinecker Jan 26 at 10:13