Is it possible to apply "standard symmetrization and density arguments" in a compact Riemannian manifold to prove something at first in a smaller subdomain of it?
So suppose we have to show finiteness of some integral, say ($f:\mathbb{R}^n\to\mathbb{R}$ is some arbitrary function)$$\int_\Omega f(x)\,dx$$ over a bounded domain $\Omega$ containing the origin in $\mathbb{R}^n$. So we can take $\Omega$ to be a ball $B(0,r)$ centered at origin with radius $r$ using symmetrization and density arguments, and then conclude the result in case of arbitrary $\Omega$. I want to know whether the same kind of argument would apply for a compact Riemannian manifold, i.e. we consider some geodesic ball $B_g(x_0,r)\subset M$ for some fixed $x_0\in M$, and then verifying the result for that geodesic ball, and then conclude for arbitrary compact $M$. Any help is appreciated.