I can't really make a sense out of my notes right now (everything went too fast in the tutoring), so I can't figure out what is actually happening here. For the sake of simplicity, assume you have the expression:
$$\int_0^{\infty} \int_{-\infty}^{zy} f(x,y) d(x, y) + \int_{-\infty}^0 \int_{zy}^{\infty} f(x, y) d(x, y).$$
Now, you want to substitute
$$v = {x \over y}$$
My notes now claim that the whole thing looks like this:
$$\int_0^{\infty} \int_{-\infty}^{z} y \ f(x,y) \ d(v y, y) + \int_{-\infty}^0 \int_{z}^{-\infty} f(vy, y) -y\ d(x, y).$$
Please don't feel confused by the last expression, it might be totally wrong, but I would like to know what the expression has to look like after the given subsitution.
Assuming I understand the notation (see below), the integral on the left of the sum seems to have a leftover $y$ from a previous step. That is, since $x =vy,$ we can integrate over $d(x,y)$ where $x \in [-\infty,zy]$ by integrating over $d(vy,y)$ where $vy \in [-\infty,zy],$ which we can do by integrating over $y\,d(v,y)$ where $v \in [-\infty,z].$ So the final result on the left of the sum might be $$\int_0^\infty \int_{-\infty}^z y f(x,y) \, d(v, y).$$ It should not be written with respect to $d(vy, y),$ since that's the same as $d(x,y)$ and there would be no justifiable reason to change the integrand or the bounds of integration. It is possible that the integrand was meant to be written $y f(vy,y)$ rather than $y f(x,y)$; they are the same thing.
The integral on the right of the final sum seems again to have the wrong differential symbol where it should have $d(v, y),$ and it also seems to have either too many negative signs or too few. If it really is just a substitution with $x=vy,$ then I would expect to see $$\int_{-\infty}^0 \int_{z}^{\infty} yf(vy, y) \,d(v, y).$$
But if the substitution for this integral is actually $x = -vy,$ then it would come out to $$\int_{-\infty}^0 \int_{-z}^{-\infty} -yf(-vy, y) \,d(v, y).$$
There are other ways this could go, depending on what further manipulations were made before getting to the result shown.
It doesn't seem that any part of the overall integration is "moved" from the right side to the left side; that is, it seems the integral on each side of the $+$ sign in the result is supposed to be equal to the integral in the same place in the initial sum.
Regarding the notation, I am guessing that $d(x,y)$ means $dx\,dy,$ that is, it indicates that the inner integral sign is with respect to $x$ and the outer integral sign is with respect to $y.$ It could also indicate integration with respect to area measure, but then I would expect a single integral sign indicating the region of integration, for example, $$\int_{(-\infty,zy]\times[0,\infty]} f(x,y)\, d(x, y).$$