Let $(X,\tau)$ be a topological vector space. Suppose that, there is a sequence of subsets $X\_n\subseteq X$ with,


1. For every $n\in \mathbb{N}$, the topology $\tau$ on $X\_n$ is second countable and metrizable space.
2. $X\_n\subseteq X\_{n+1}$ and $X=\bigcup X\_n$.


Q. Is the Borel sigma algebra coming from the weak topology $\sigma(X,X^\*)$ is the same as the Borel sigma algebra coning from $\tau$?



Borel sigma algebra coming from the weak topology on TVS
There are separable metric TVS $X$ whose topological dual is trivial (one example is $L\_p([0,1])$ for $0\le p<1$). Such $X$ satisfies 1 and 2 (w.r.to the sequence $X\_n:=X$), and the two sigma algebras are of course not the same.


