I was wondering about the sense of the following definition:
Let (M,g) be a Riemmanian manifold and $\varphi\in C^{\infty}(M)$ a smooth function, such that $\varphi\geq 0$ or $supp(\varphi)\subset M$ is compact. Let $A\subset M$ be mesuarable. If there is a chart $x:U\rightarrow\mathbb{R}^n$ with $\overline{A}\subset U$, then the following makes sense (since the right hand side is integrable, i.e. less than infinity):
$\int\limits_{M}\varphi dV:=\int\limits_{x(A)}(\varphi \circ x^{-1})\cdot \sqrt{det(g_ij)}dV$
I don't know why the right hand side is integrable if $\varphi\geq 0$. If the support is compact, then of course we have $\int\|\|<\infty$ But what is the explanation in the other case?
Thank you a lot.