This is a followup to:

Strictly Positive Measures on Countable Boolean Algebras

Suppose a countable Boolean algebra B is a subalgebra of the power set of the reals. (For example, let B be the Boolean algebra of definable sets of reals, in one of the various natural senses of definability.)

A strictly positive, continuous measure on B is a function m from B to [0,1] such that

(i) $m(b)=0$ iff $b=0$,

(ii) $m(1)=1$,

(iii) $m(a+b)=m(a)+m(b)$ whenever $a$ and $b$ are disjoint.

(iv) if $A_1$, $A_2$, $A_3$ ... is an increasing countable sequence of elements of B, and the union $A=\bigcup_{i=0}^{\infty}$ is also in $B$, then $m(A)=\lim_{i \rightarrow \infty} m(A_i)$

Is there a strictly positive, continuous measure on every such countable Boolean Algebra?