Let $X$ denote the space of bounded Borel functions $f\colon [0,1] \to \mathbb{R}$. Let $M$ denote the space of finite Borel measures on $[0,1]$. What is the *largest* family $F \subset M$ such that for any $f\in X$ there exists a sequence $(f_n)_{n=1}^\infty \subset C[0,1]$ such that for any $\mu \in F$
$$
\int_{0}^{1} f_n(t) \, d\mu(t) \to \int_{0}^{1} f(t) \, d\mu(t) \quad as \quad n\to \infty,
$$
i.e. $f_n \to f$ in the topology $\sigma(X,F)$.

In other words, what is the

largestfamily $F \subset M$ that $C[0,1]$ issequentiallydense in $(X, \sigma(X, F)))$?

Clearly $F$ includes all absolutely continuous (with respect to Lebesgue) measures, because any bounded Borel function can be approximated with uniformly bounded continuous functions converging strongly in $L^1$. On the other hand $F\not = M$ as pointed out in a related post (which is actually a motivation for this question).

*Update.* If we asked $C[0,1]$ be dense (not sequentially dense) in $(X, \sigma(X,F))$, then one could have taken simply $F=M$. Indeed, for any finite set of measures $\mu_1$, ..., $\mu_n$ from $M$ we know that $C[0,1]$ is dense in $L^1([0,1], |\mu_1| + \ldots + |\mu_n|)$. But since we are asking $C[0,1]$ to be sequentially dense in $(X, \sigma(X,F))$, even existence of the *largest* family $F$ is not clear, as noted in the comment by @ChristianRemling . So existence of $F$ can be considered as a part of the question.

*Another update.* As pointed out in the comments by @ChristianRemling the term *largest* is ambiguous. So let me clarify this point. Let $\mathcal F$ denote the collection of all families $F \subset M$ such that $C[0,1]$ is sequentially dense in $(X, \sigma(X,F))$. By *largest* I meant maximal in the partially ordered set $(\mathcal F, \subset)$. Now a more precise version of my question has 3 parts:

- Does there exist a maximal element of $\mathcal F$?
- Does there exist a maximal element of $\mathcal F$ which is a superset of the family of all absolutely continuous measures?
- Can any of these maximal elements (if exists) be described explicitly?

sequentialdensity (added some details to the question). $\endgroup$ – Skeeve Mar 14 '19 at 17:46