# What is the measure-theoretic definition of the condition probability?

Let $$(S, \mathcal{S}, \Pr)$$ be a probability space. Let $$(\mathcal{X}, \mathcal{B})$$ be a standard Borel space and a measurable map $$X : S \to \mathcal{X}$$. Let $$(\Omega, \tau)$$ be a standard Borel space and a measurable map $$\Theta : S \to \Omega$$. Then

$$\mu_{\Theta \mid X}(A \mid x) = \int_A \frac{f_{X \mid \Theta}(x \mid \theta)}{\int_\Omega f_{X \mid \Theta}(x \mid t) \, d\mu_\Theta(t)} \, d\mu_\Theta(\theta)$$ for all $$A \in \tau$$ and $$x \in \mathcal{X}$$. I have seen the above equation in Theory of statistics (Schervish, 2005) and in this post.

What I don't understand is that a measure deals with sets but $$x$$ is not a set here.

Does $$A|x$$ represent a set here?

Shouldn't the conditional measure $$\mu_{\Theta \mid X}$$ be a measure on a set difference $$A|B$$ where $$A \in \tau,B \in \mathcal{B}$$?

• Conditional probability is defined in terms of the conditional expectation of the indicator variable. For the measure-theoretic account of that, please see stats.stackexchange.com/questions/230545.
– whuber
Aug 17, 2022 at 16:28
• @whuber, I don't think this answers why we use $A|x$ instead of $A|B$. Aug 17, 2022 at 16:38
• I had hoped it would be clear that the conditional expectation is a function.
– whuber
Aug 17, 2022 at 17:00
• @whuber, I think OP is asking whether $x$ here is a single value or a set of values since measure functions are defined on subsets and not on single values.
– gbd
Aug 17, 2022 at 17:08
• @gbd Once again, conditional expectations are functions. Although not all the symbols in that integral are defined in this post, it is evident $f$ is some kind of a density function, whence $x$ does not designate a set: it is manifestly an element of $\Omega.$ $A$ explicitly is a set (it's in the sigma algebra $\tau$).
– whuber
Aug 17, 2022 at 17:25