The complementation of an additive measure up to $\sigma$-additivity by means of an extension of the space

For an algebra $\mathfrak A$ of subsets of a set X there is constructed a set $\widetilde X\supset X$ and an algebra of its subsets so that the mapping $\widetilde A\to A=\mathop\mathfrak A\limits^\sim\cap A$ is a one-to-one correspondence between $\mathop\mathfrak A\limits^\sim$ and $\mathfrak A$ and for each additive measure $M$ on $\mathfrak A$ the measure $\widetilde\mu$ on $\mathop\mathfrak A\limits^\sim$ defined by the equation $\widetilde\mu(\widetilde A)=\mu(A)$ is countably additive.