Much has been said of the $dx$ notation used for integration on this site, but some writers of mathematics papers (especially physicists), write integrals as
$$ \int dxf(x) $$
For instance, one way of writing the law of total probability I have seen is
$$ f_Y(y) = \int_{-\infty}^{\infty}dxf_{X|Y}(x|y)f_Y(y) $$
What ideas are the authors trying to emphasize through this notation?