Uniform integrabblity and the lebesgue theorem on convergence in $L_0$-valued measures

Abstract

We study integrals $∫fdμ$ of real functions over $L_0$-valued measures. We give a definition of convergence of real functions in quasimeasure and, as a special case, in $L_0$-measure. For these types of convergence, we establish conditions of convergence in probability for integrals over $L_0$-valued measures, which are analogous to the conditions of uniform integrability and to the Lebesgue theorem.