Borel sigma algebra coming from the weak topology on TVS

Question

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 $\tau$?

Answer 1

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.