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!

• You can't condition on measure zero sets, it just doesn't make sense. Related question – icurays1 Jan 25 '13 at 21:38
• I don't get it? You want a measure theoretc treatment of Bayesian statistics? – Michael Greinecker Jan 25 '13 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 '13 at 22:40
• @MichaelGreinecker To send somebody to that reference for "a treatment of Bayesian statistics"? Honestly, this is rather flabbergasting. – Did Jan 26 '13 at 3:20
• @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 '13 at 10:13