## Integration with respect to general random measures in the Riemann sense.(English. Ukrainian original)Zbl 1004.60050

Theory Probab. Math. Stat. 62, 127-131 (2001); translation from Teor. Jmovirn. Mat. Stat. 62, 119-123 (2000).
Let $$X$$ be an arbitrary precompact metric space, let $$\mathcal B$$ be a Borel $$\sigma$$-algebra of subsets of $$X,$$ and let $$(\Omega,{\mathcal F},P)$$ be a probability space. By definition, a random measure $$\mu$$ is a family of random variables $$\{\mu(A); A\in \mathcal B\}$$ such that $$\mu(A_{n})@> P >> 0$$ as $$A_{n}\to\emptyset$$, and $$\mu(A\cup B)=\mu(A)+\mu(B)$$ a.s. for $$A\cap B=\emptyset.$$ There are no restrictions of nonnegativity and existence of the moments for the measure $$\mu.$$ In this sense it means that $$\mu$$ is a general random measure. The integrals of the form $$\int_{A}gd\mu$$, where $$g$$ is a real measurable function on $$X,$$ $$\mu$$ is a random measure, $$A\in\mathcal B$$ were introduced by V. N. Radchenko [“Integrals of general stochastic measures” (1999; Zbl 0953.60035)]. The aim of this paper is to study the integrals of the form $$\int_{A}gd\mu$$ with random function $$g(x,\omega)=\sum_{k=1}^{\infty}\xi_{k}(\omega)g_{k}(x),$$ where $$\xi_{k}$$ are random variables on $$(\Omega,{\mathcal F},P),$$ $$g_{k}:X\to R$$ are measurable functions with $$|g_{k}(x)|\leq 1.$$ The conditions of integrability of the random integrands over the random measure in the Riemann sense are proposed.

### MSC:

 60G57 Random measures 28D99 Measure-theoretic ergodic theory 60H05 Stochastic integrals

### Keywords:

general random measure; integration in Riemann sense

Zbl 0953.60035