Непрерывные меры [Итоговый научный отчет по междисциплинарному интеграционному проекту СО РАН: “Разработка теории и вычислительной технологии решения обратных и экстремальных задач с приложением в математической физике и гравимагниторазведке”]

We consider the binary algebra $(B,+,\cdot)$ and abel semigroup $(H,+)$ with neutral $0$-elements and with topologies ensuring a continuity of additive and multiplicative transfers. Let $A$ – subalgebra of algebra $B$. We shall name an additive and continuous mapping $m\colon A\to H$ as an abstract measure. One of fundamental tasks of the general measure theory is an investigation of existance conditions of a continuous extention $\bar m\colon\bar A\to H$ of measure $m\colon A\to H$ to its definition domain closure $\bar A $. A number of author works mentioned in literature is dedicated to solving this problem. There is their brief review fnd some new results described in the article. These results are connected with extentions of a vector measure to an integral sum and the last one to an integral.