It is known that the probability density function $f(x)$ and the cumulative distribution function $F(x)$ are related as $f(x) = \frac{\partial F(x)}{\partial x}$. However I am confused why at some places the density function is written as just $dF(x)$.
This came up in the definition of Stieltjes Transform: $m(z) = \int \frac{1}{x - z} dF(x)$. And it is mentioned that
The density function $f(x) := dF(x)$ in the distributional sense
Is this just the issue with notation or is there specific reason to write the density function as $dF(x)$?