Suppose $(\Omega, \mathcal{\Sigma})$ is a measurable space equipped with two probability measures $P_1$ and $P_2$. $\mathcal{F}$ is a field in $\Omega$ and $\sigma(\mathcal{F})=\mathcal{\Sigma}$. (Added: A field of sets is defined to be closed under finite union and complement and contain $\Omega$.)
I was wondering if $\lim_{\delta \rightarrow 0} \quad \sup_{B \in \mathcal{F}, P_2(B) < \delta} P_1(B) = 0$ implies $\lim_{\delta \rightarrow 0} \quad \sup_{B \in \mathcal{\Sigma}, P_2(B) < \delta} P_1(B) =0$ and, if yes, what conclusions or theorems can be used to prove it?
Thanks in advance!