Convergence of integrals of unbounded real functions in random measures
Teoriâ veroâtnostej i ee primeneniâ, Tome 42 (1997) no. 2, pp. 358-364
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$\sigma$-additive random measures and integrals with respect to them of real valued functions are considered in the most general setting. The statement of convergence of $\int f d\mu_n\stackrel{\mathsf{P}}{\longrightarrow}\int f d\mu$, $n\to\infty$, is proved under conditions similar to uniform integrability. An analogue of the Valle–Poussin theorem is established. A criterion is given for the relation $\int f_ng d\mu\stackrel{\mathsf{P}}{\longrightarrow}\int g d\eta$, $n\to\infty$, to hold for all bounded $g$.
Keywords:
random measure, $L_0$-valued measure, integral with respect torandom measure, uniform integrability, Valle–Poussin theorem.
@article{TVP_1997_42_2_a9,
author = {V. M. Radchenko},
title = {Convergence of integrals of unbounded real functions in random measures},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {358--364},
publisher = {mathdoc},
volume = {42},
number = {2},
year = {1997},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_1997_42_2_a9/}
}
V. M. Radchenko. Convergence of integrals of unbounded real functions in random measures. Teoriâ veroâtnostej i ee primeneniâ, Tome 42 (1997) no. 2, pp. 358-364. http://geodesic.mathdoc.fr/item/TVP_1997_42_2_a9/