This is a slightly soft question, but I am wondering if it is sensible to talk about the moments of a given measure $\mu$. In particular, suppose that $X$ is a random variable on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Is there a way to define $\mathbb{E}[\mu]$ for measures $\mu$ on $\mathbb{R}$ such that $\mathbb{E}[\mu_X] = \mathbb{E}_\mathbb{P}[X]$? Seeing as the expectation of a random variable is only dependent on its distribution, I am thinking that such a definition should be possible without reference to the underlying random variable.
If this definition is possible, can we then extend it to $n$th moments of a given measure? Could we also extend it (using the Bochner Integral) to arbitrary measures $\mu: B \to [0,\infty]$ where $B$ is a Banach space?