Is there a generalized form of the differentiable change of variables theorem for Lebesgue integrals? That is, if we consider the well known change of variables theorem: If $\phi : X \rightarrow X$ is a diffeomorphism of open sets in $\mathbb{R}^n$, $X \subseteq \mathbb{R}^n$ is measurable, and $f : X \rightarrow \mathbb{R}$ is measurable, then:
$ \int_X f(y) dy = \int_X f(\phi(x)) d(\phi(x)) = \int_X f(\phi(x)) |\det D\phi(x)| dx $
I'de like to weight between some countable set of (simple) diffeomorphic mappings instead of just one, that is, let $\Phi = \{\phi_i \ | \ i \in \mathbb{N}, |\det D\phi_i(x)| = 1\}$. Additionally, I'de like to weight between these transformations as a convex-combination, so I define a weighting function, $w : X \times \mathbb{N} \rightarrow \mathbb{R}$, where: $\sum_i w(\phi_i^{-1}(y),i)$ = 1. Then, I'de like to show:
$ \int_X \sum_i w(x,i) f(\phi_i(x)) dx = \int_X f(x) dx $
My attempt at a proof is: \begin{align} \int_X \sum_i w(x,i) f(\phi_i(x)) dx & = \sum_i \int_X w(x,i) f(\phi_i(x)) dx\\ & = \sum_i \int_X w(x,i) f(\phi_i(x)) |\det D\phi_i(x)|dx\\ & = \sum_i \int_X w(\phi_i^{-1}(y),i) f(y) dy\\ & = \int_X f(y) \sum_i w(\phi_i^{-1}(y),i) dy \\ & = \int_X f(y) dy \end{align}
Trouble is, I'm not all that familiar with measure theory and I would need to show that my weighting function is measurable in order to invoke the single-mapping change of variables theorem mentioned above. Perhaps I cannot do this without being more explicit about what this function actually is, but at the same time, it's just a simple weight vector, normalized in some unique way, over a countable set it would be nice if I could say something at this level of generality. Also, perhaps it makes more sense to start at the case where $\Phi$ is finite, which is fine, but I have the same issues with the proof with this assumption.