The expectation of a geometric measure of a level set of a random function. (English. Russian original) Zbl 0629.60017

Radchenko, A. N.

Theory Probab. Math. Stat. 32, 109-112 (1986); translation from Teor. Veroyatn. Mat. Stat. 32, 99-102 (1985).

Suppose that (\(\Omega\),\(\sigma\),P) is a complete probability space and \(\xi\) : \(X\times \Omega \to Y\) is a random function, a.s. continuous on \(X\subset R^ m\), where Y is an arbitrary metric space. Let \(S^ k_ y(B)=S^ k(x\in B:\xi (x,\omega)=y)\) be the k-dimensional spherical measure of a level set and N(B) be the number of points in a set B, \(N_ y(p,z,B)=N(x\in B:\xi (x,\omega)=y\), \(p(x)=z)\), p is a projector from \(R^ m\) on \(R^ k\). The results of the paper are the following two theorems:
Theorem I. For any \(\sigma\)-compact set \(A\subset X\), \(t\in [1,+\infty)\) and \(k=I,...,m\) \(E[S^ k_ y(A)]^ t\cdot \beta^ t_ t(m,k)\geq \int_{O^*(m,k)}dp[\int_{R^ k}EN_ y(p,z,A)dz]^ t.\)For \(t=I\) equality holds for some \(A\subset X\) and \(k\in \{I,...,m\}\) if (and only if, in the case when \(S^ k_ y(A)<\infty\) a.s.) the level set \(\{\) \(x\in A:\xi (x,\omega)=y\}\) is a.s. countably \((S^ k,k)\)-rectifiable. Further, \[ ES^ k_ y(A)=\frac{1}{\beta_ 1(m,k)}\iint EN_ y(p,z,A)dpdz, \] where \[ \beta_ 1(m,k)=[\Gamma (\frac{k+1}{2})\cdot \Gamma (\frac{m-k+1}{2})]/[\Gamma (\frac{m+1}{2})\cdot \Gamma ()] \] and E denotes expectation with respect to P.
Theorem 2: Let for some \(k\geq I\) and \(y\in Y\) the variable \(EN_ y(p,z,.)\) be translation-invariant and finite on compact sets for almost all \(p\in O^*(m,k)\) and \(z\in R^ k\). Then for these k and y and for any \(\sigma\)-compact set \(A\subset X\) and any \(t\geq I\) \[ E[S^ k_ y(A)]^ t\geq [\frac{L^ m(A)}{\beta_ t(m,k)}]^ t\cdot \int_{O^*(m,k)}[N_ y(P)]^ tdp. \] For \(t=I\) equality holds for some \(A\subset X\) if (and only if, in the case when \(S^ k_ y(A)<\infty\) a.s.) the set \(\{\) \(x\in A:\xi (x,\omega)=y\}\) is a.s. countably \((S^ k,k)\)-rectifiable. Further, for any Borel set A’\(\subset A\) \(ES^ k_ y(A')=S^ k_ y\cdot L^ m(A'),\)where \[ S^ k_ y=[\beta_ 1(m,k)]^{-1}\cdot \int_{O^*(m,k)}N_ y(p)dp, \] and \(L^ m(.)\) is m-dimensional Lebesgue measure. For related work see the author, ibid. 31, 116-125 (1984; Zbl 0589.60043); English translation in Theory Probab. Math. Stat. 31, 131-140 (1985).

Reviewer: V.K.Oganyan

MSC:

60D05	Geometric probability and stochastic geometry
53C65	Integral geometry

Keywords:

geometric measure; random function; spherical measure of a level set; translation-invariant and finite on compact sets

Citations:

Zbl 0589.60043