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

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

Zbl 0589.60043