Favard measure
of order of a subset
in a Euclidean space
of dimension
A generalization of the Hausdorff measure; it was introduced by J. Favard [1]. The precise definition is: The group of motions of induces on the collection of its
-dimensional affine subspaces a left-invariant Haar measure
that is unique up to normalization, and the set
induces a function
the value of which at an affine subspace
is the number of points in the intersection
. The Favard measure of
is the value of
at
, if the normalizing constant is chosen so that
for the
-dimensional unit cube
.
The Favard measure of order of a set
does not exceed its Hausdorff
-measure
and, in the case when
, it is the same as
if and only if
splits into a countable number of parts one of which has Hausdorff
-measure zero and each of the others can be situated on a smooth
-dimensional manifold.
References
[1] | J. Favard, "Une définition de la longueur et de l'aire" C.R. Acad. Sci. Paris , 194 (1932) pp. 344–346 |
[2] | H. Federer, "Geometric measure theory" , Springer (1969) |
Favard measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Favard_measure&oldid=13597