Shifting integration variables

Question

I'm not sure how to pose this question precisely, but I'll try. I'm trying to see what happens when you have an integral of the form $\int \mathrm{d}x \,f(x-g(z))$ and you try and write it as $\int \mathrm{d}\mu(h)\, f(h)$ where $h=x-g(z)$ and $\mathrm{d}\mu$ is the measure for the new variable. This is just a shift of variables, but since $g$ isn't constant I'm not sure how the measure changes. Is there a general procedure for this?