Integrals of pullbacks and the Inverse function theorem(s?)

Question

The usual story goes like this:

Smooth picture (?):

For a smooth bijection $\phi: M \to N$ between $n$-manifolds the following is true:

1. $\phi^{-1}$ is a local diffeomorphism a.e.

2. Given an open set $U \subset N$ and a form $\omega \in \Omega^k(U)$ we have the equality: $\int_U \omega= \int_{\phi^{-1}(U)} \phi^*\omega$.

Satisfied with this simple and very intuitive picture I slowly came to believe that this is the most general change of variables theorem there could be. That is, until I met the following theorem in a measure theoretic context.

Theorem 1:

Let $U\subset\mathbb{R}^n$ be a measurable subset and $\phi :U\to \mathbb{R}^n$ be injective and differentiable.

$\implies$ $\int_{\phi(U)} f = \int_U f\circ\phi |\det D\phi|$ for all real valued functions $f$.

Then upon searching the internet I came by a weaker version of the inverse function theorem for everywhere differentiable functions:

Theorem 2:

Let $U \subset \mathbb{R}^n$ and $f:U \to \mathbb{R}^n$ be differentiable s.t. $Df_{x_0}$ has full rank for all $x_0 \in U$
$\implies$ $f$ is a local differentiable homeomorphism.

Which leads to the following generalization:

Differentiable picture (conjecture): For a differentiable bijection $\phi: M \to N$ between $n$-manifolds the following is true:

1. $\phi^{-1}$ is a local differentiable homeomorphism a.e.

2. Given an open set $U \subset N$ and a form $\omega \in \Omega^k(U)$ we have the equality: $\int_U \omega= \int_{\phi^{-1}(U)} \phi^*\omega$.

In search of a unifying picture i listed the properties a function $\phi$ must have so that the pullback wont change the value of the integral.

I. $\phi$ must be locally absolutely continuous. (otherwise it could send a null set to a positive measure set). This also establishes the almost everywhere differentiability of $\phi$.

II. $D \phi$ must have full rank almost everywhere.

If we can preform change of variables with $\phi$. What more properties must it have?

Following the connection with lipschitz functions i arrived at the following unifying conjecture:

The great conjecture:

For a locally lipschitz bijection $\phi: M \to N$ between $n$-manifolds the following is true:

1. $\phi^{-1} $ is locally bi-lipschitz (a.e.?)

2. Given an open set $U \subset N$ and a form $\omega \in \Omega^k(U)$ we have the equality: $\int_U \omega= \int_{\phi^{-1}(U)} \phi^*\omega$.

Is this true?

Answer 1

If you consider continuous injections (resp. homeomorphisms onto their range) instead of locally Lipschitz bijections (resp. locally bi-Lipschitz), then the modified conjecture is true because of Brouwer's theorem on invariance of domain, with the proviso that in (2) you should consider $n$-forms instead of $k$-forms because the domain $U$ of integration is assumed open and hence is $n$-dimensional. Formula (2) in this case is simply the change of variables formula for integrals with respect to a measure $\mu$ and its pushforward $\phi_*\mu$, once you write $n$-forms in terms of the volume measure associated to a Riemannian metric on $M$ (recall that a continuous map is always Borel measurable). Here I'm assuming $M$ and $N$ oriented for simplicity.

Otherwise (assuming back your original hypotheses), you should consider locally $k$-rectifiable subsets $U$ of the target manifold $N$ in (2) instead of $U$ open ("locally" equals "countably" if your manifolds are second countable). All that remains is to show that the inverse of $\phi$ is locally Lipschitz. This, however, needs an additional hypothesis as follows. By Rademacher's theorem, any locally Lipschitz map $\phi:M\rightarrow N$ is differentiable almost everywhere; more precisely, the (Clarke sub)differential $D\phi$ of $\phi$ is a locally bounded set-valued map which is single-valued almost everywhere. That being said, the additional hypothesis you need in your conjecture to settle part (1) is that $D\phi$ takes only non-singular values, for in this case you can invoke Clarke's inverse function theorem for Lipschitz maps (Theorem 1 of F.H. Clarke, "On The Inverse Function Theorem". Pac. J. Math. 64 (1976) 97-102). Part (2) follows immediately by using the $k$-dimensional Hausdorff measure with respect to some Riemannian metric on $M$.

Some of the comments addressed the possibility of doing away with the hypothesis of $D\phi$ being non-singular by invoking Sard's theorem. However, this does not work with so little regularity assumed from $\phi$ - a necessary hypothesis for Sard's theorem to hold true, as pointed in Sard's original paper ("The Measure of Critical Values of Differentiable Maps", Bull. Amer. Math. Soc. 48 (1942) 883-890) is that $\phi$ should be at least $\mathscr{C}^1$. $\phi$ being locally Lipschitz is not good enough - as shown by J. Borwein and X. Wang ("Lipschitz functions with maximal subdifferentials are generic", Proc. Amer. Math. Soc. 128 (2000) 3221–3229), the set of 1-Lipschitz functions $f$ such that all points in its domain are critical (in the sense that $Df$ contains zero everywhere) is generic with respect to the uniform topology. In particular, generically one cannot expect the inverse of a locally Lipschitz bijection, albeit certainly continuous, to be locally Lipschitz (even if only up to a subset of measure zero).