In the Conditional expectation Wikipedia page, in the Classical definition: Conditional expectation with respect to an event section, there is a $P(dx | H)$ notation, whose meaning I don't understand:
According to the explanation below it should be $P(dx\cap H)/P(H)$, but the $dx$ part is still confusing.
What I want is a very clear, purely real analysis notation, i.e., one that specifies the domain over which to integrate, the corresponding $\sigma$-algebra, the measure and the integrand function, instead of the sorta elusive probabilistic formulation. It seems the integral domain is $\mathcal X$, but neither the $\sigma$-algebra nor the measure is readily apparent to me. Any help?
Implicit in that explanation is 1) some state space $(\cal{X},\mathscr{X})$ in which $X$ takes values, and 2) that $P(\cdot\mid H)=P(\cdot,H)$ is a probability measure on $(\cal{X},\mathscr{X})$. The existence of a $P(\cdot,H)$ satisfying 2) and also satisfying the relation mentioned toward the end of the screenshot $P(A,H)=P(A|H)=P(A\cap H)/P(H)$ (also implicit is $A\in\mathscr{X}$) amounts to the existence of a regular conditional probability distribution for $X$ on $(\Omega,\mathcal{F},P)$, which will hold, e.g., when $X$ is real-valued. As pointed out in the comment below, when we pass to the more everyday bayesian formula definition of event $A$ conditional on event $B$ we actually want to write $P(X^{-1}(A)\cap H)/P(H)$
"What I want is a very clear, purely real analysis notation, i.e., one that specifies the domain over which to integrate," --> $\cal X$
the corresponding σσ-algebra, --> $\mathscr X$
the measure --> $P(\cdot,H)$
and the integrand function" --> the identity
Maybe some of the confusion is also due to the notation of the integrator $P(dx,H)$. This just means integrate with dummy variable $x$ using the measure $P(\cdot,H)$, e.g., for lebesgue measure I might write $\int_0^\pi cos(x)\lambda(dx)$ for $\int_0^\pi cos(x)dx$.
Another possible point of confusion: using the same letter $P$ to refer to the underlying measure on the probability space $(\Omega,\cal{F},P)$ and the rcpd $P(\cdot | \cdot)$.