View source for Favard measure

''of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038280/f0382801.png" /> of a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038280/f0382802.png" /> in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038280/f0382803.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038280/f0382804.png" />''

A generalization of the [[Hausdorff measure|Hausdorff measure]]; it was introduced by J. Favard [[#References|[1]]]. The precise definition is: The group of motions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038280/f0382805.png" /> induces on the collection of its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038280/f0382806.png" />-dimensional affine subspaces a left-invariant [[Haar measure|Haar measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038280/f0382807.png" /> that is unique up to normalization, and the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038280/f0382808.png" /> induces a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038280/f0382809.png" /> the value of which at an affine subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038280/f03828010.png" /> is the number of points in the intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038280/f03828011.png" />. The Favard measure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038280/f03828012.png" /> is the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038280/f03828013.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038280/f03828014.png" />, if the normalizing constant is chosen so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038280/f03828015.png" /> for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038280/f03828016.png" />-dimensional unit cube <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038280/f03828017.png" />.

The Favard measure of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038280/f03828018.png" /> of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038280/f03828019.png" /> does not exceed its Hausdorff <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038280/f03828020.png" />-measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038280/f03828021.png" /> and, in the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038280/f03828022.png" />, it is the same as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038280/f03828023.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038280/f03828024.png" /> splits into a countable number of parts one of which has Hausdorff <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038280/f03828025.png" />-measure zero and each of the others can be situated on a smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038280/f03828026.png" />-dimensional manifold.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Favard,   "Une définition de la longueur et de l'aire"  ''C.R. Acad. Sci. Paris'' , '''194'''  (1932)  pp. 344–346</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Federer,   "Geometric measure theory" , Springer  (1969)</TD></TR></table>