Some Properties of Complete Boolean Algebras

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.

Keywords:

q^prime-compactness , Strongly algebraically closed algebras , Complete Boolean algebras