Let $X$ be a set.

Let $\mathcal{R}$ be a set of subsets of $X$ such that

$\{\} \in \mathcal{R}$

and

For all members $A$ and $B$ of $\mathcal{R}$, $\;\; (A\cup B)-(A\cap B) \; \in \; \mathcal{R} \;\;$.

and

For all sequences $\; \langle A_0,A_1,A_2,A_3,...\rangle \;$ of members of $\mathcal{R}$, $\;\;\;\; \displaystyle\bigcap_{n=0}^{\infty} \; A_n \;\; \in \;\; \mathcal{R}$

.

Let $\;\; \phi : \mathcal{R} \to \mathbb{R} \;\;$ be such that for all pairwise disjoint

sequences $\; \langle A_0,A_1,A_2,A_3,...\rangle \;$ of members of $\mathcal{R}$,

if $\;\;\;\; \displaystyle\bigcup_{n=0}^{\infty} \; A_n \;\; \in \;\; \mathcal{R} \;\;\;\;$ then $\;\;\;\; \displaystyle\sum_{n=0}^{\infty} \; \phi(A_n) \;\; = \;\; \phi\left(\displaystyle\bigcup_{n=0}^{\infty} \; A_n\right) \;\;\;\;$.

Does it follow that there exists a member $P$ of $\mathcal{R}$ such that

for all members $A$ of $\mathcal{R}$, $\;\; A\cap P \; \in \; \mathcal{R} \;\;$ and $\; \phi(A-P\hspace{.01 in}) \leq 0 \leq \phi(A\cap P\hspace{.01 in}) \;\;$?