Consider the integral of a function $f$ over a domain $D \subset \mathbb{R}$, and a function (automorphism?) $g: D \rightarrow D$ with $g'(x) > 0$ for all $x \in D$. Then we may write
$$ \int_D f(x) \mathrm{d} x = \int_D \overbrace{f(g(y)) g'(y)}^{h(y)} \mathrm{d} y = \int_D h(y) \mathrm{d} y. $$
In other words, by applying the transformation $g$, we can effectively replace the integrand but keep the domain of integration. This has some interesting applications. For example, with a well-designed $g$ we can replace certain singular integrands with smooth replacements (essentially locally contracting and expanding the domain). My motivation here is numerical integration, in which case this transform would let us use a standard quadrature rule for smooth integrands on $D$.
I assume this is a well-known concept. Does it have a particular name? Can anyone please point me to literature, or otherwise give examples where these kinds of integral transforms are useful? I'm particularly interested in construction of suitable $g$ for different problems, especially in the multivariate case.