Generalization of Vitali-Hahn-Saks theorem for algebras

Question

By Vitali-Hahn-Sack theorem, we know that, if $(\mathbb{P}_n)_{n\in\mathbb{N}}$ is a sequence of probabilities defined on a $\sigma$-algebra $\mathcal{F}$ of subset of a set $\Omega$, such that for each $ F \in \mathcal{F}$ the sequence $(\mathbb{P}_n(F))_{n\in\mathbb{N}}$ converges, then the limit function is a measure on $\mathcal{F}$. I'm wondering if the same result holds true if $\mathcal{F}$ is just an algebra of subsets of $\Omega$ and $(\mathbb{P}_n)_{n\in\mathbb{N}}$ is a sequence of pre-measures defined on $\mathcal F$ with total mass 1, i.e. if also the limit function is a pre-measure. Can anyone give me an answer or provide any reference? Thanks in advance.

Answer 1

What follows is a counterexample.

Let $\mu_C$ be the counting measure on $\mathbb{N}$. Define $$\mathcal{F}:=\{F\in2^\mathbb{N}\ |\ (\mu _C(F)<+\infty)\lor(\mu _C(\mathbb{N}\backslash F)<+\infty)\}.$$ Clearly $\mathcal{F}$ is an algebra of subsets of $\mathbb{N}$. Define $$\forall n\in\mathbb{N},\forall F\in\mathcal{F}, \mathbb{P}_n(F):=\chi_F(n)=\delta_n(F).$$ Then for each $n\in\mathbb{N}$ it is clear that $\mathbb{P}_n$ is a pre-measure of mass $1$ defined on $\mathcal{F}$. Now, $$\forall F\in\mathcal{F}, \mathbb{P}_n(F)\rightarrow\mathbb{P}(F):=\begin{cases} 0\ \ \textrm{if} \ \ \mu_C(F)<+\infty\\ 1\ \ \textrm{if} \ \ \mu_C(F)=+\infty \end{cases}, n\rightarrow+\infty.$$ However $\mathbb{P}$ isn't a pre-measure because, for example, $\sum_{n\in\mathbb{N}}\mathbb{P}(\{n\})=0\neq1=\mathbb{P}(\mathbb{N})$.