Some Properties of Complete Boolean Algebras

Abstract

The main result of this paper is a characterization of the strongly algebraically closed algebras in the  lattice of all real-valued continuous functions and the equivalence classes of $lambda$-measurable. We shall provide conditions  which strongly algebraically closed algebras carry a strictly positive Maharam submeasure. Particularly, it is proved that if $B$ is a strongly algebraically closed lattice  and $(B,, sigma)$ is a Hausdorff space  and $B$ satisfies  the   $G_sigma$ property, then $B$ carries a strictly positive Maharam submeasure.