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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 $dF(x)$?

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1 Answer

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I think it's a notation thing. Maybe like the notation for a differential operator.

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