Measure on Boolean algebra

Question

my question is:

Suppose that $\mathfrak{B}$ is a measurable Boolean algebra, does this mean that "Every measure on $\mathfrak{B}$ should be strictly positive ? or this will be the case after QUOTIENTING it.

Please someone explain this ?

Thanks in advance

Answer 1

If you do not require that the measure be strictly positive, then every Boolean algebra admits a two valued measure (Boolean prime ideal theorem). Also not all Boolean algebras admit strictly positive measure. There is a nice characterization in an old paper of Kelly which you can access here.