derivative in $\mathbb{P}$ (interpretation)

Question

Can someone give me an interpretation of the following notation of a probability?

$\mathbb{P} (X\in \mathrm{d}x)$ with the usual conventions.

Answer 1

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. $\Bbb P_X(A) = \Bbb P (X\in A).$