In a paper I am reading, it is seemingly suggested that, if $\nu(dx)$ is a Lévy measure, then the following holds for a function $f(x)$ which is smooth (and satsifies some integrability conditions): $$ \int^{-x}_{-\infty}f(x)\,\nu(\mathrm dy) + \int^0_{-\infty}[f(y)-f(x)]\, \nu(\mathrm dy-x) = \int^{-x}_{-\infty}f(x+y)\, \nu(\mathrm dy) $$ I would like to verify this equation but I am not sure how.
I think my main problem is understanding what $\nu(\mathrm dy-x)$ means. I understand the basic behaviour of a Lévy measure, but I have not seen a case like this before. I suspect it might have something to do with convolutions of measures? I would appreciate if someone could verify this before I potentially set out on learning about them, and it would be fantastic if it also can be shown how it works in the context of that equation.
Source of equation: paper. The equation can be obtained after some manipulations of the expression on p 11, under "Step 4".