Rationale behind this notation for integrations?

Question

The marginal distribution formula from Wikipedia is: $$ p_X(x) = \int_y p_{X,Y} (x, y) dy $$ It makes sense.

But some use the following notation: $$ \mathbb{P}_{X} = \int_Y d\mathbb{P}_{XY} $$ Why do they use $d\mathbb{P}_{XY}$ there, where $\frac{d\mathbb{P}_{XY}}{dy} \neq \mathbb{P}_{XY}$? What is the rationale behind the notation?

Answer 1

A univariate distribution is often denoted with cdf $P$ and pdf $p$ so $pdx=dP$. Similarly, for multivariate distributions $p_{X,\,Y}dxdy=d\Bbb P_{XY}$ and $\Bbb P_X=\int p_X dX=\int_{xy}p_{X,\,Y}dxdy=\int_{xy}d\Bbb P_{XY}$.