What does the $F(dy)$ in $\int_{yWhat does the $F(dy)$ in
$$f(x)=\int_{y<x}g(y)F(dy)$$
means?
Note that $F$ is a distribution function/probability measure.
Furthermore, how to use Cauchy-Schwarz to get the below:
$$
|f(x)|^2\leq F(x)\int g^2 dF
$$
I came across this from a proof in Huber's "Robust Statistics". The picture contains full details of the proof.
--- Edit ---
If someone could verify Masacroso's answer, it would be great.



As I said in the comments this is just a notation to specify the variable of integration, that is, notations like $dF, dF(y), F(dy)$ are exactly the same thing. Indeed it can be seen because in the cited text it says that $f$ is the density of $F$ and $g=f'/f$, and this agree with $F(dy)=f(y) dy$ as
$$
\int_{y<x}g(y)F(dy)=\int_{y<x}\frac{f'(y)}{f(y)}f(y)dy=\int_{y<x}f'(y)dy=f(x)
$$
