Can anyone explain the concept of regular conditional probability? Can anyone explain regular conditional expectation and give me some intuition? I understand every term used by the definition, but still do not get what it is trying to say.
Also the following integral is from wikipedia. 
$$P{\big (}A\cap T^{{-1}}(B){\big )}=\int _{B}\nu (x,A)\,P{\big (}T^{{-1}}(dx){\big )}.$$ 
What is the integral used here? Is it Lebesgue? But doesnt lebesgue integral have a measure after d, so it is sth like $\int f dP$ where P is a measure?
Or is it lebesgue stieltjes integral? But doesnt lebesgue stieltjes have a distribution function after d? so it is sth like $\int f dQ$ where Q is a distribution function?
 A: It helps if $\nu(x,A)$ was denoted as $\nu_A(x)$. Then, if we had a random variable $X$, we could easier interpret $$\nu_A(x)\ \text{ as }\ P(A\mid X=x).$$ So $\nu_A(x) $ is a measure depending on $x$. The problem with this definition is that $P(A\mid X=x)$ is usually not defined when $P(X=x)=0$. This is because, according to the classical definition
$$P(A\mid X=x)=\frac{P(A\cap X=x)}{P(X=x)}$$
and we don't like the $0$ in the denominator. But $P(A\cap X=x)$ is also $0$, so we may try to define the conditional probability, even in these cases, as a limit...

For example, let $X$ be uniformly distributed over $[0,1]$, and let $A=\big\{X<\frac12\big\}$. Then 
$$\nu_{\{X<1/2\}}(x)=\lim_{\Delta\to 0}\frac{P(X<\frac12\cap x\leq X\leq x+\Delta)}{P(x\leq X\leq x+\Delta)}=\begin{cases}1&\text{ if }&0\leq x<\frac12\\
0&\text{ otherwise. }\end{cases}$$
Take a measurable set, say $B=\left[\frac14, \frac34\right]$. What is $$\int_B\nu_{\{X<1/2\}}(x)Q(dx)?$$
where, yes, $Q$ is the distribution of $X$, that is, the question can be asked again: What is
$$\int_{\frac14}^{\frac34}\nu_{\{X<1/2\}}(x)dx=\int_{\frac14}^{\frac12}dx=\frac14?$$
Indeed, with $T=X$, $$\frac14=P(A\cap T^{-1}(B))=P\left(X<\frac12\cap \frac14<X<\frac34\right)=P\left(\frac14<X<\frac12\right).$$
So, $\nu_{\{X<1/2\}}(x)$ behaves as if it was $P(X<\frac12\mid X=x).$
