Let $X$ be a topological vector space. Let us say that $X$ has property **P** if there exists a sequence of closed subsets $\{X_n\}$ such that

1- $X=\bigcup X_n$

2- The relative topology is both metrizable and second countable on $X_n$'s.

Q. Assume $X$ satisfying **P** property. Let $m: X\times X\to X$ be an associative multiplication. Is $m$ measurable?