# Measurability of the product on particular topological vector spaces

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?

• Do you assume $m$ bilinear? – Pietro Majer Apr 21 '18 at 13:38
• @ Pietro Majer Absolutely yes. – Ali Bagheri Apr 21 '18 at 13:45

The answer seems to be "no" even for metrizable separable Banach spaces, which have property $\mathbf P$.
Take any infinite-dimensional separable Banach space $X$, fix a non-zero point $x_0\in X$ and a discontinuous linear functional $f:X\to\mathbb R$ such that $f(x_0)=1$. It is well-known that $f$ is not measurable (in many possible senses).
Define the bilinear function $m:X\times X\to X$ by $m(x,y)=f(x)\cdot f(y)\cdot x_0$. It can be show that the function $m$ is associative. The non-measurability of $f$ implies the non-measurability of the bilinear form $m$.