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''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" />''
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{{MSC|28A}}
  
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" />.
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[[Category:Classical measure theory]]
  
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.
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{{TEX|done}}
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The term Favard measures denote a family of [[Outer measure|outer measures]] in the Euclidean space and the corresponding [[Measure|measures]] (when restricted on their respective [[Algebra of sets|$\sigma$-algebras]] of measurable sets). They are often called ''integralgeometric measures''. Some special cases were considered for the first time by Favard in {{Cite|Fa}}.
 +
 
 +
====Definition for $p=1$====
 +
First of all consider  $1\leq m<n$ integers. Consider
 +
* the [[Orthogonal group|orthogonal group]] $O(n)$ of linear isometries of $\mathbb R^n$ and the [[Haar measure]] $\theta_n$ on it;
 +
* the [[Grassmann manifold|Grassmannian]] $G_{m,n}$ of (unoriented) $m$-dimensional planes of $\mathbb R^n$; for any element $V\in G_{m,n}$ we let $p_V: {\mathbb R}^n\to V$ be the orthogonal projection;
 +
* the measure $\gamma_{m,n}$ on $G_{m,n}$ given by
 +
\[
 +
\gamma_{m,n} (A) = \theta \left(\left\{g\in O(n): g (V)\in A\right\}\right)
 +
\qquad \mbox{for all Borel $A\subset G_{m,n}$}\, ,
 +
\]
 +
where $V$ is any element of $G_{m,n}$;
 +
* the [[Hausdorff measure|Hausdorff $\alpha$-dimensional measures]] $\mathcal{H}^\alpha$ on $\mathbb R^n$.
 +
 
 +
'''Definition 1'''
 +
If $E\subset \mathbb R^n$ is a [[Borel set]] the value of the Favard measure (with parameter $p=1$) on $E$ is given by
 +
\[
 +
\mathcal{I}^m_1 (E) := \int_{G_{m,n}} \int_V \mathcal{H}^0 \left(E \cap p_V^{-1} (\{a\}\right)\, d\mathcal{H}^m (a)\, d\gamma_{m,n} (V)\, .
 +
\]
 +
 
 +
Cp. with Section 5.14 of {{Cite|Ma}}.
 +
 
 +
====Definition for general $p$: Caratheodory construction====
 +
For $p\in [1, \infty]$ it is possible to define outer measures $\mathcal{I}^m_p$. We start by definining the [[Set function]] $\zeta^m_p$ on the Borel $\sigma$-algebra $\mathcal{B}$. For $p<\infty$ we set
 +
\[
 +
\zeta^m_p (B) := \left(\int_{G_{m,n}} \left(\mathcal{H}^m (p_V (B))\right)^pd\gamma_{m,n} (V)\right)^{\frac{1}{p}}
 +
\]
 +
whereas we define
 +
\[
 +
\zeta^m_\infty (B) = {\rm ess sup}\, \left\{ \mathcal{H}^m (p_V (B)): V\in G_{m,n}\right\}\, .
 +
\]
 +
Note that the $\gamma_{m,n}$-measurability of the map $V\mapsto \mathcal{H}^m (p_V(B))$ is a subtle issue (see Section 2.10.5 of {{Cite|Fe}}).
 +
 
 +
We next follow the usual [[Outer measure|Caratheodory construction]] of outer measures.
 +
 
 +
'''Definition 2'''
 +
Let $\delta \in ]0, \infty]$, $p\in [1, \infty]$ and $A\subset \mathbb R^n$. We set
 +
\[
 +
\mathcal{I}^m_{p,\delta} (A) = \inf \left\{\sum_{i=0}^\infty \zeta^m_p (B_i): B_i \in \mathcal{B}, {\rm diam}\, (B_i)<\delta \;\mbox{and}\; B \subset \bigcup_i B_i \right\}\, .
 +
\]
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The function $\delta\mapsto \mathcal{I}^m_{p,\delta} (A)$ is nonincreasing and we therefore define
 +
\[
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\mathcal{I}^m_p (A) := \beta_p (n,m)^{-1}\; \lim_{\delta\downarrow 0}\; \mathcal{I}^m_{p, \delta} (A)\, .
 +
\]
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The normalizing factor $\beta_p (n,m)$ is chosen in such a way that $\mathcal{I}^m_p (B)$ coincides with $\mathcal{H}^m (B)$ when $B$ is the unit box in an $m$-dimensional plane $V$.
 +
 
 +
====Properties====
 +
* The outer measures in '''Definition 2''' satisfy [[Outer measure|Caratheodory's criterion]] and hence the Borel sets are $\mathcal{I}^m_p$-measurable.
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* On the Borel $\sigma$-algebra the measure $\mathcal{I}^m_1$ as in '''Definition 1''' coincides with the one of '''Definition 2''' (cp. with Theorem 2.10.15 of {{Cite|Fe}}; for general $p$'s there is a suitable inequality).
 +
* The measures $\mathcal{I}^m_p$ coincide all with the Hausdorff measure $\mathcal{H}^m$ on smooth $m$-dimensional submanifolds of $\mathbb R^n$ and, more in general, on [[Rectifiable set|rectifiable subsets]] of dimension $m$ (Cp. with Section 3.2.26 of {{Cite|Fe}})).
 +
* For any $A$, $p\mapsto \beta_p (n,m)\, \mathcal{I}^m_p (A)$ is nondecresing.
 +
* In {{Cite|Ma2}} Mattila constructed a compact set $A\subset \mathbb R^2$ such that $\mathcal{I}^1_1 (A) < \mathcal{I}^1_p (A) = \infty$ for every $p>1$.
  
 
====References====
 
====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>
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{|
 +
|-
 +
|valign="top"|{{Ref|Fa}}|| J. Favard,  "Une définition de la longueur et de l'aire"  ''C.R. Acad. Sci. Paris'' , '''194'''  (1932)  pp. 344-346
 +
|-
 +
|valign="top"|{{Ref|Fe}}||      H. Federer, "Geometric measure    theory". Volume 153 of Die    Grundlehren  der mathematischen    Wissenschaften. Springer-Verlag New    York Inc., New  York, 1969.    {{MR|0257325}} {{ZBL|0874.49001}}
 +
|-
 +
|valign="top"|{{Ref|Ma}}||        P. Mattila, "Geometry of sets and  measures in Euclidean spaces.    Fractals and rectifiability".      Cambridge Studies in Advanced    Mathematics, 44. Cambridge University      Press, Cambridge,  1995.    {{MR|1333890}} {{ZBL|0911.28005}}
 +
|-
 +
|valign="top"|{{Ref|Ma2}}|| P. Mattila, "An example illustrating integralgeometric measures", ''Amer. J. Math.'' '''108''' (1986) pp. 693-702.
 +
|-
 +
|}

Revision as of 17:38, 5 October 2012

2020 Mathematics Subject Classification: Primary: 28A [MSN][ZBL]

The term Favard measures denote a family of outer measures in the Euclidean space and the corresponding measures (when restricted on their respective $\sigma$-algebras of measurable sets). They are often called integralgeometric measures. Some special cases were considered for the first time by Favard in [Fa].

Definition for $p=1$

First of all consider $1\leq m<n$ integers. Consider

  • the orthogonal group $O(n)$ of linear isometries of $\mathbb R^n$ and the Haar measure $\theta_n$ on it;
  • the Grassmannian $G_{m,n}$ of (unoriented) $m$-dimensional planes of $\mathbb R^n$; for any element $V\in G_{m,n}$ we let $p_V: {\mathbb R}^n\to V$ be the orthogonal projection;
  • the measure $\gamma_{m,n}$ on $G_{m,n}$ given by

\[ \gamma_{m,n} (A) = \theta \left(\left\{g\in O(n): g (V)\in A\right\}\right) \qquad \mbox{for all Borel '"`UNIQ-MathJax14-QINU`"'}\, , \] where $V$ is any element of $G_{m,n}$;

Definition 1 If $E\subset \mathbb R^n$ is a Borel set the value of the Favard measure (with parameter $p=1$) on $E$ is given by \[ \mathcal{I}^m_1 (E) := \int_{G_{m,n}} \int_V \mathcal{H}^0 \left(E \cap p_V^{-1} (\{a\}\right)\, d\mathcal{H}^m (a)\, d\gamma_{m,n} (V)\, . \]

Cp. with Section 5.14 of [Ma].

Definition for general $p$: Caratheodory construction

For $p\in [1, \infty]$ it is possible to define outer measures $\mathcal{I}^m_p$. We start by definining the Set function $\zeta^m_p$ on the Borel $\sigma$-algebra $\mathcal{B}$. For $p<\infty$ we set \[ \zeta^m_p (B) := \left(\int_{G_{m,n}} \left(\mathcal{H}^m (p_V (B))\right)^pd\gamma_{m,n} (V)\right)^{\frac{1}{p}} \] whereas we define \[ \zeta^m_\infty (B) = {\rm ess sup}\, \left\{ \mathcal{H}^m (p_V (B)): V\in G_{m,n}\right\}\, . \] Note that the $\gamma_{m,n}$-measurability of the map $V\mapsto \mathcal{H}^m (p_V(B))$ is a subtle issue (see Section 2.10.5 of [Fe]).

We next follow the usual Caratheodory construction of outer measures.

Definition 2 Let $\delta \in ]0, \infty]$, $p\in [1, \infty]$ and $A\subset \mathbb R^n$. We set \[ \mathcal{I}^m_{p,\delta} (A) = \inf \left\{\sum_{i=0}^\infty \zeta^m_p (B_i): B_i \in \mathcal{B}, {\rm diam}\, (B_i)<\delta \;\mbox{and}\; B \subset \bigcup_i B_i \right\}\, . \] The function $\delta\mapsto \mathcal{I}^m_{p,\delta} (A)$ is nonincreasing and we therefore define \[ \mathcal{I}^m_p (A) := \beta_p (n,m)^{-1}\; \lim_{\delta\downarrow 0}\; \mathcal{I}^m_{p, \delta} (A)\, . \] The normalizing factor $\beta_p (n,m)$ is chosen in such a way that $\mathcal{I}^m_p (B)$ coincides with $\mathcal{H}^m (B)$ when $B$ is the unit box in an $m$-dimensional plane $V$.

Properties

  • The outer measures in Definition 2 satisfy Caratheodory's criterion and hence the Borel sets are $\mathcal{I}^m_p$-measurable.
  • On the Borel $\sigma$-algebra the measure $\mathcal{I}^m_1$ as in Definition 1 coincides with the one of Definition 2 (cp. with Theorem 2.10.15 of [Fe]; for general $p$'s there is a suitable inequality).
  • The measures $\mathcal{I}^m_p$ coincide all with the Hausdorff measure $\mathcal{H}^m$ on smooth $m$-dimensional submanifolds of $\mathbb R^n$ and, more in general, on rectifiable subsets of dimension $m$ (Cp. with Section 3.2.26 of [Fe])).
  • For any $A$, $p\mapsto \beta_p (n,m)\, \mathcal{I}^m_p (A)$ is nondecresing.
  • In [Ma2] Mattila constructed a compact set $A\subset \mathbb R^2$ such that $\mathcal{I}^1_1 (A) < \mathcal{I}^1_p (A) = \infty$ for every $p>1$.

References

[Fa] J. Favard, "Une définition de la longueur et de l'aire" C.R. Acad. Sci. Paris , 194 (1932) pp. 344-346
[Fe] H. Federer, "Geometric measure theory". Volume 153 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag New York Inc., New York, 1969. MR0257325 Zbl 0874.49001
[Ma] P. Mattila, "Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability". Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. MR1333890 Zbl 0911.28005
[Ma2] P. Mattila, "An example illustrating integralgeometric measures", Amer. J. Math. 108 (1986) pp. 693-702.
How to Cite This Entry:
Favard measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Favard_measure&oldid=28317
This article was adapted from an original article by L.D. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article