If $m(B):=\int_Bf(x)dV_g(x)$ for the Riemannian volume form $V_g$ and $f\in L^1(M,dV_g)$, which subset $B$ of $M$ are $m$-measurable?

Question

Let $(M,g)$ be a compact Riemannian manifold and take $f\in L^1(M, dV_g)$ for the Riemannian volume form $V_g$. If we define $m(B):=\int_Bf(x)dV_g(x)$, which sets are $m$-measurable? Heck, maybe the real questions is that what sets are $dV_g$-measurable? I'm inclined to believe that the Riemannian volume form behaves like a Lebesgue measure, so e.g. all open subsets of $M$ are $dV_g$-measurable. But I have had hard time finding a reference which considers measures on Riemannian manifolds.