When I try to model diffusely defined objects, I frequently found myself in the need to refer to a measure in a way independent of the dimensional attributes of the space of definition of the objects.
In other words: I try to find a single word that be in a general sense, a synonym of {"volume", "surface","lenght", etc} regardless of the dimension of the space. That is: the Lebesgue Measure of the object.
Without having found any convention, I use sometimes the word "lebesguian" for that, letting "size" for referring in a vague way, to the "like-diameter" metrics that do not depend of the dimensionallity.
So you could express diffuse laws on the beliefs of the model (for instance the dimensionallity of the space ) as for instance:
"The lebesguian of an object varies more or less like its size to the power of its dimensionallity"
I wonder if this would be valid, or there is a better and/or more popular words for saying those things?
If the "intrinsic" dimension of an object $O$ is $d\geq0$ then its $d'$-dimensional Hausdorff measure is $=\infty$ when $d'<d$, and is $=0$ when $d'>d$. If you are lucky its $d$-dimensional Hausdorff measure is a finite number $\mu>0$, which you then could call the intrinsic volume of $O$. But you cannot compare these volumes for different dimensional objects $O$, $O'$, since they scale differently under linear scaling.