Defining measure

As mentioned already, it is straightforward to extend a measure on ${\cal C}$ to $X$ if the topology associated with the phase variables is trivial. It may not be possible, however, to obtain an invariant measure. In most cases, ${\cal C}$ is a transformation group, in which the Haar measure exists, thereby yielding the ``true'' volume in a sense that is not sensitive to parameterizations of ${\cal C}$. This was observed for $SO(3)$ in Section 5.1.4. For a general state space $X$, a Haar measure may not exist. If a Riemannian metric is defined, then intrinsic notions of surface integration and volume exist [133]; however, these may be difficult to exploit in a sampling-based planning algorithm.