The following integral is from wikipedia. 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?
What is this?
$$P{\big (}A\cap T^{{-1}}(B){\big )}=\int _{B}\nu (x,A)\,P{\big (}T^{{-1}}(dx){\big )}.$$
$P(T^{-1}(dx))$ is alternative notation for $dT_*P(x)$, where $T_*P$ is the pushforward measure of $P$ along $T$. So they mean Lebesgue integral here.
In general $\mu(dx)$ is alternative notation for $d\mu(x)$.