Let $X$ be a topological vector space. Assume that there exists a sequence of finite range measurable functions $\phi_n:X\to X$ with $\lim\phi_n(x)=x$.
Q. Can we concluded that $X$ is hereditery Lindelof?
By an interesting example Taras Banakh proved that: $X$ is not necessarily written by countable union of second countable subsets (see here).