It's not at all obvious to me how "assigning a probability measure" translates into an actual algorithm. For one thing, if you have an n-D manifold embedded in an (n+1)-D space, any set of points on the manifold will have measure zero in the enclosing region.
To give an intuitive definition, consider the volume near the n-D manifold, up to a distance d. As you get nearer the volume goes down it starts looking like ~A*2d -- in fact we define A as the limit as d goes to 0 of the volume divided by 2d. You can extend this to k-D manifolds on n-D spaces by normalizing this volume with a n-k dimensional ball.
Now you can partition this set in any way you want and apply the rule I mentioned for each partition (p=measure(Partition)/measure(Manifold)). Simply because that is precisely how one might define "equal probabilities", there exists an algorithm which partitions the manifold into simply connected compact sets of decreasing measure which converges to giving "equal probabilities": the definition implies an algorithm.
Edit: panic above clarified the requirement. I missed that that measure(Manifold) must of course exist, but I'm confident all else is self-evident.
