A conjecture about generating algebras on a probability space Suppose that $(X,\mathscr F,\mathbb P)$ is a probability space. Let $\mathscr E\subseteq\mathscr F$ be an algebra (i.e., it is a non-empty collection closed under complementation and finite unions) that generates the $\sigma$-algebra $\mathscr F$: $\sigma(\mathscr E)=\mathscr F$.
Let $F\in\mathscr F$. Is it true that for any $\varepsilon>0$, there exists some $E\in\mathscr E$ such that $$|\mathbb P(E)-\mathbb P(F)|<\varepsilon?$$ One might try considering the outer measure generated by $\mathbb P|\mathscr E$, but an exact proof or counterexample has eluded me.
Thank you for any input in advance.

ADDED: I had tried using Dynkin’s lemma first, but I was unable to prove that the subcollection of such sets in $\mathscr F$ that satisfy the desired property was a Dynkin system (in particular, I couldn’t directly show that the subcollection was closed under taking nested differences). However, the answer by user24142 helped me realize that another generating-class argument, based on the monotone-class lemma, easily implies the desired result.
 A: Your conjecture is correct. The reason is that the set of sets that you can approximate within $\mathscr{F}$ is a $\sigma$-algebra. That its closed under complements should be clear, and the countable union condition isn't too hard. It therefore is a $\sigma$-algebra that contains $\mathscr E$ and is contained by $\mathscr F$, so it must be $\mathscr F$.
A: This is an elaboration of the (slightly simplified version of the) answer suggested by user24142.
Let $$\mathscr S\equiv\{F\in\mathscr F\,|\,\forall\varepsilon>0,\exists E\in\mathscr E:|\mathbb P(E)-\mathbb P(F)|<\varepsilon\}.$$ Clearly, $\mathscr E\subseteq\mathscr S$. I claim that $\mathscr S$ is a monotone class. Fix $\varepsilon>0$ and suppose that $(F_m)_{m\in\mathbb N}$ is a non-decreasing collection of sets in $\mathscr S$; let $F$ denote their union. Then, since $\lim_{m\to\infty}\mathbb P(F_m)=\mathbb P(F)$, there exists some $n\in\mathbb N$ such that $|\mathbb P(F_n)-\mathbb P(F)|<\varepsilon/2$. For such an $F_n$, choose $E\in\mathscr E$ such that $|\mathbb P(E)-\mathbb P(F_n)|<\varepsilon/2$. Then, $|\mathbb P(E)-\mathbb P(F)|<\varepsilon$. It follows that $F\in\mathscr S$. A similar argument shows that $\mathscr S$ is closed under countable non-increasing intersections. It follows that $\mathscr S$ is a monotone class, indeed, so the monotone-class lemma (remember that $\mathscr E$ is an algebra) implies that $\mathscr F=\sigma(\mathscr E) \subseteq \mathscr S$, as desired.
