Favard measure

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

The term Favard measures denotes 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

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

• the 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 [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.