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

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

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    It's best understood as just notation: you are integrating in the variable $x$ against the law of the random variable $X$. Notational issues like this are just part of life with Lebesgue integration of functions of several variables (since you need to say both which variable is which and which measure is used for which variable). – 2017-01-23

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

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    And a perfectly rigorous alternative is $$E(g(X))=\int_\mathbb Rg(x)dP_X(x)$$ where $P_X$ denote the image measure of $P$ by $X$, aka the distribution of $X$. – 2017-01-23