derivative in $\mathbb{P}$ (interpretation) Can someone give me an interpretation of the following notation of a probability?
$\mathbb{P} (X\in \mathrm{d}x)$ with the usual conventions.
 A: Usually if you want to integrate a measurable function $f$ with respect to the distribution of a random variable $X$ you write
$$
\int_\Bbb R f(x) \;\Bbb P (X \in dx).
$$
For example if you want to calculate the expected value of $X$ you can use this notation:
$$
\Bbb E [X]= \int_\Bbb R x \Bbb P (X \in dx).
$$
Very often you can circumvent this notation (I never use it):


*

*If $F$ is the distribution function of $X$ then $\Bbb E [X] = \int_\Bbb R x \;  dF(x)$ 

*If you denote the measure induced by $X$ with $\mu$ then  $\Bbb E [X] = \int_\Bbb R x \;  d\mu(x)$.

*If $X$ has a pdf $f$ you can write $\Bbb E [X] = \int_\Bbb R x  f(x)\;dx$ instead.


So there are some alternatives.  $\Bbb P (X \in dx)$ is most likely used if you do not want to introduce any further notation (like $F,\mu, f$).
Edit:
As Did said, there is an alternative which works without introducing any notation: 
$$
\Bbb E [X] = \int_\Bbb R x \;  d\Bbb P_X(x),
$$
where $P_X$ is the push-forward measure (or image measure), i.e. $\Bbb P_X(A) =  \Bbb P (X\in A).$
