View source for Favard measure

{{MSC|28A}}

[[Category:Classical measure theory]]

{{TEX|done}}

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
\[
\mathcal{I}^m_{p,\delta} (A) = \inf \left\{\sum_{i=0}^\infty \zeta^m_p (B_i): B_i \in \mathcal{B}, {\rm diam}\, (B_i)<\delta \;\mbox{and}\; B \subset \bigcup_i B_i \right\}\, .
\]
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 [[Outer measure|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 {{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====
{|
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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.
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