Vestnik of Samara University. Natural Science SeriesVestnik of Samara University. Natural Science Series2541-75252712-8954Samara National Research University571010.18287/2541-7525-2017-23-4-33-39UnknownTHE BROOKS-JEVETT THEOREM ON UNIFORM DIMENTRICULARITY ON A NON-SIGMA-FULL CLASS OF SETSSribnayaT. A.morenov.sv@ssau.ruSamara National Research University2512201723433392501201825012018Copyright © 2017, Sribnaya T.A.2017<p>For a sequence of exhaustive composition-triangular set functions defined on a non-sigma-complete class of sets, more general than the ring of sets, the Brooks-Jewett theorem on uniform exhaustibility is proved. As a corollary, we have obtained analogue of the Brooks-Jewett theorem for functions defined on a sigma-summable class of sets. It is shown that if, in addition to the property compositional triangularity, the set functions have the composite semi-additivity property and are continuous from above at zero, then an analog of Nikodyms theorem on equicontinuous weak continuity is valid for them. The corresponding results are obtained for a family of quasi-Lipschitz set functions.</p>композиционно-треугольные функции множества, композиционно-полуаддитивные функции множества, не-сигма-полный класс множеств, мультипликативный класс множеств, исчерпываемость, непрерывность сверху в нуле, равномерная исчерпываемостьcomposition-triangular set functions, composition-semi-additive set functions, non-sigmacomplete class of sets, multiplicative class of sets, exhaustibility, continuity from above at zero, uniform exhaustibility[[1] Dunford N., Schwartz J. Lineinye operatory. Obshchaia teoriia [Linear operators. Part 1: General theory]. М.: IIL, 1962, 896 p. [in Russian].][[2] Brooks J.K., Jewett R.S. On finitely additive vector measures. Proc. Nat. Acad. Sci USA, 1970, Vol. 67, no 3, pp. 1294–1298 [in English].][[3] Guselnikov N.S. O teoremakh Bruksa-Dzhevetta i Nikodima [On the theorems of Brooks-Jewett and Nicodemus]. In: Sb. Teoriia funktsii i funkts. analiz [Collection Theory of functions and functional analysis]. L., 1975, pp. 45–54 [in Russian].][[4] Klimkin V.M. Vvedenie v teoriiu funktsii mnozhestva [Introduction to the theory of set functions]. Izdatel’stvo Saratovskogo universiteta, Kuibyshevskii filial, 1989, 210 p. [in Russian].][[5] Molto A. On the Vitali-Hahn-Saks theorem. Proc. Royal. Soc.. Edinburgh, 1981, Sect. A90, pp. 163–173 [in English].][[6] Schachermayer W. On some classical measure-theoretic theorems for non-sigma-complete Boolean algebras. Dissertationes Math.. Warszawa, 1982, Vol. 214, pp. 1–33 [in English].][[7] Friniche F.I. The Vitali-Hahn-Saks theorem for Boolean algebras with the subsequential interprolation property. Proc. Amer. Math. Soc., 1984, Vol. 92, no 3, pp. 362–366 [in English].][[8] Lucia P., Marales P. Some Consequences of the Brooks-Jewett theorem for Additive Uniform Semigroup-valued Functions. Conf. Semin. Mat. Univ.. Bari, 1988, Vol. 227, pp. 1–23 [in English].][[9] Guselnikov N.S. O prodolzhenii kvazilipshitsevykh funktsii mnozhestva [On the extension of quasi-Lipschitz set functions]. Matem. zametki [Mathematical Notes], 1975, Vol. 17, no. 1, pp. 21–31 [in Russian].][[10] Seever G.L. Measures on F-spaces. Trans. Amer. Math. Soc., 1968, Vol. 133, pp. 267–280 [in Russian].][[11] Sribnaya T.A. Kriterii ravnomernoi ischerpyvaemosti semeistva vektornykh vneshnikh mer [A criterion for the uniform exhaustibility of a family of vector external measures]. Vestnik Samarskogo gosuniversiteta. Estestvennonauchnaia seriia [Vestnik of Samara State University], 2012, no 6 (97), pp. 58–65 [in Russian].]