Definition of conditional probabiliy as function dependent on $\sigma$-Algebra I know that for events $A,B$ with $P(B) > 0$ the conditional probability is defined as
$$
 P(A | B) = \frac{P(A \cap B)}{P(B)}. 
$$
Of course by regarding $A$ as constant, and varying $B$ we get a function $P(A | \cdot)$ by
$B \mapsto P(A | B)$, and so we get a function $P(A | \{ \cdot \}) : \Omega \to \mathbb [0,1]$ by $\omega \in \Omega \mapsto P(A | \{ \omega \})$ (assuming each $\omega$ has a non-zero probability). Is this function a random variable? Guess not because there is no measure space given on $[0,1]$ for a measure (like $P$).
Also I stumble in the way conditional probabilities for $\sigma$-Algebra are defined. For this let $\mathcal F$ be an $\sigma$-Algebra, then the conditional probability $P(A | \mathcal F)$ is a $\mathcal F$-measurable and integrable random variable such that
$$
 \int_G P(A | \mathcal F) d P = P(A \cap G)
$$
for all $G \in \mathcal F$. This makes no sense to me, why now a function. In the classical definition I got a number, which could be interpreted as the probability of an event given another event, but here I have a collection of events I depend on, and the conditional probability is a function... makes no sense to me?
Has my construction above something to do with the way conditional probabilites for $\sigma$-Algebras are defined? I just tried to come from the classical definition to the new one...
 A: You are certainly not alone in asking this question to yourself: at least that seemed very strange and non-intuitive to me when I came across it. More to say, this definition of conditional probability is the Doob's version of the one proposed by Kolmogorov. None of them were immediately happily accepted by the mathematics community at the time they were invented. Let's take a closer look.
Whenever you define conditional probability given that some event happened, it makes sense to use the familiar ratio definition. In such case the conditional probability is a single value. Now, if you want to compute probability conditional that some random variable $\xi$ took value $x$, that shall be a function of $x$. When Kolmogorov thought of that problem, the Radon-Nikodym theorem have just been proved in its general form, so it appeared to him that it's the right tool to define conditional probabilities. We can now see that this definition indeed satisfies the properties we'd expect, however I think it's very much worth reading his original exposition in "Foundation of ..." There he first discusses the discrete case before generalizing it, so that should be similar to your thoughts.
