Let $X$ be a set, let $B(X)$ denote the vector space of bounded real functions on $X$, and let $V\subseteq B(X)$ be a vector subspace containing $1$ and closed under monotone limits.
Let now $\Sigma_V$ be the set of subsets of $X$ such that their indicator function is in $V$, and suppose that $\Sigma_V$ is a $\sigma$-algebra (on the nose). Is it true that all functions of $V$ are $\Sigma_V$-measurable?
If not, do we need to assume any additional conditions?
Edit: This is not the (classic) monotone class theorem. I'm not asking whether the $\Sigma_V$-measurable functions are in $V$, but rather the opposite: I'm asking whether all the functions in $V$ are measurable for $\Sigma_V$.
Try this one: Let $X$ be $[0,1]$ and let $V$ be the set $\{a+bx : a,b\in \mathbb R\}$.
Additional conditions... Maybe it is enough to add
$f,g \in V \Longrightarrow f\vee g \in V$.