Suppose $M$ is a smooth manifold and $g$ be any continuous $2$-tensor field. Then we define the integration of any compactly supported continuous function on $M$ as the following.
If $\text{supp}f\subseteq U$ where $(U,\phi)$ is a smooth chart, define $\int_Mf:=\int_{\phi(U)}f\circ\phi^{-1}\sqrt{|\text{det}(g_{ij})|}dx_1\dots dx_n $ where $g_{ij}:=g(\partial/\partial x_i,\partial/\partial x_i)$. One can see that this definition is independent of the chosen smooth chart. Now for general compactly supported function we can define the integration using partition of unity.
Is the above mentioned construction alright? In literature, I have seen the above mentioned construction when $g$ is a non-degenerate symmetric $2$-tensor field. Why one insists $g$ to be symmetric and non-degenerate, when we can define the integration with respect to any $2$-tensor field?