Given a smooth manifold $M$, there are several ways of constructing measures on $M$. The most common procedure I've seen is by starting with a $(0,2)$ tensor field $T$ on $M$, and defining for each chart $(U,x)$, the function $\rho_x := \sqrt{|\det T_{(x), ij}|}$. These functions then give us a non-negative scalar density, $\rho$ on $M$. Using this scalar density, we can essentially (chart by chart) "pull back" the Lebesgue measure to get a well-defined positive measure $\lambda_{\rho}$ on $M$. For example if we're on a (pseudo)-Riemannian manifold, we can use the metric tensor to get the usual Riemannian volume measure. If we're on a symplectic manifold, we can use this same recipe with the symplectic form $\omega$ (or equivalently we just take $\left|\frac{\omega^n}{n!}\right|$... where we think of a scalar density as a section of an appropriate bundle).
Now, my question is, can someone provide me some interesting examples where other types of measures naturally arise; for example are there some other structures which are studied (aside from Riemannian/symplectic, since these are the only two I know) in more advanced areas of geometry/analysis from which a natural notion of a positive measure on a manifold arises. Also, could you provide a (brief) explanation of where such a construction is used/why it is useful. I'm mainly asking to just broaden my perspective. Thanks in advance.