What does $$\intop_{D}\rho(x)dx=\intop_{D}\rho(x)\phi(x)dx$$ imply about $\phi(x)$ for some positive definite function $\rho(x)$ which satisfies $$\intop_{D}\rho(x)dx=M$$
For some nonzero M. $\phi(x)$ and $\rho(x)$ are both continuous, smooth, nontrivial (i.e. $\phi,\rho\neq1$ ) I have an argument/proof I'll post later to the effect that:
$$\intop_{D}\rho(x)dx=\intop_{D}\rho(x)\phi(x)dx\quad\Longrightarrow\quad\intop_{D}\mid\phi(x)\mid^{2}dx=1$$ However; I'm not 100% sure about it, would appreciate any thoughts. thank you!
EDIT
As I've learned more about the math behind this problem, it has become evident to me that this problem is more deep than I had initially considered. I had tried to give a “watered down” version (ie the one above) but I think I need to get more into the mathematical detail behind this:
I was considering compact 4-dimensional (pseudo) Riemannian metric spaces without boundary such that the volume of the space is given by:
$$V=\intop_{M}\sqrt{|g|}d^{4}x$$
What I've come to realize is that I'm looking for the set of local conformal transformations $\tilde{g}=u(x)^{2}g$ such that the volume of the manifold is unchanged:
$$\intop_{M}\sqrt{|g|}d^{4}x=\intop_{M}\sqrt{|\tilde{g}|}d^{4}x=\intop_{M}u(x)^{4}\sqrt{|g|}d^{4}x$$
I had initially “posed this question for a 1-dimensional space” in my initial question. I've further come to realize that the ricci curvature of the space for $g$ needs to be constant whereas for $\tilde{g}$ it is not constant.
This then ties the problem immediately to the well-known Yamabe problem (though I have only just realized this). Essentially my question is are the above conditions sufficient to ensure there exists such a function $u$? What are the conditions placed upon $u$? (such as the attempted integration condition I originally tried to place) Note I study physics and have been a bit “out of my depth” on this problem.
If $\rho$ an arbitrary continuous function in an open inverval, then: $$\int \rho(x)(1-\phi(x))dx = 0 \Rightarrow 1-\phi(x)=0 $$ Find out here (a general case): https://en.wikipedia.org/wiki/Fundamental_lemma_of_calculus_of_variations